Reed–Solomon error correction

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id="toc-Data_storage" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Data_storage"> <div class="vector-toc-text"> 2.1 Data storage </div> </a> <ul id="toc-Data_storage-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bar_code" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bar_code"> <div class="vector-toc-text"> 2.2 Bar code </div> </a> <ul id="toc-Bar_code-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Data_transmission" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Data_transmission"> <div class="vector-toc-text"> 2.3 Data transmission </div> </a> <ul id="toc-Data_transmission-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Space_transmission" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Space_transmission"> <div class="vector-toc-text"> 2.4 Space transmission </div> </a> <ul id="toc-Space_transmission-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Constructions_(encoding)" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Constructions_(encoding)"> <div class="vector-toc-text"> 3 Constructions (encoding) </div> </a> <button aria-controls="toc-Constructions_(encoding)-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Constructions (encoding) subsection </button> <ul id="toc-Constructions_(encoding)-sublist" class="vector-toc-list"> <li id="toc-Reed_&_Solomon's_original_view:_The_codeword_as_a_sequence_of_values" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Reed_&_Solomon's_original_view:_The_codeword_as_a_sequence_of_values"> <div class="vector-toc-text"> 3.1 Reed & Solomon's original view: The codeword as a sequence of values </div> </a> <ul id="toc-Reed_&_Solomon's_original_view:_The_codeword_as_a_sequence_of_values-sublist" class="vector-toc-list"> <li id="toc-Simple_encoding_procedure:_The_message_as_a_sequence_of_coefficients" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Simple_encoding_procedure:_The_message_as_a_sequence_of_coefficients"> <div class="vector-toc-text"> 3.1.1 Simple encoding procedure: The message as a sequence of coefficients </div> </a> <ul id="toc-Simple_encoding_procedure:_The_message_as_a_sequence_of_coefficients-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Systematic_encoding_procedure:_The_message_as_an_initial_sequence_of_values" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Systematic_encoding_procedure:_The_message_as_an_initial_sequence_of_values"> <div class="vector-toc-text"> 3.1.2 Systematic encoding procedure: The message as an initial sequence of values </div> </a> <ul id="toc-Systematic_encoding_procedure:_The_message_as_an_initial_sequence_of_values-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Discrete_Fourier_transform_and_its_inverse" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Discrete_Fourier_transform_and_its_inverse"> <div class="vector-toc-text"> 3.1.3 Discrete Fourier transform and its inverse </div> </a> <ul id="toc-Discrete_Fourier_transform_and_its_inverse-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-The_BCH_view:_The_codeword_as_a_sequence_of_coefficients" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_BCH_view:_The_codeword_as_a_sequence_of_coefficients"> <div class="vector-toc-text"> 3.2 The BCH view: The codeword as a sequence of coefficients </div> </a> <ul id="toc-The_BCH_view:_The_codeword_as_a_sequence_of_coefficients-sublist" class="vector-toc-list"> <li id="toc-Systematic_encoding_procedure" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Systematic_encoding_procedure"> <div class="vector-toc-text"> 3.2.1 Systematic encoding procedure </div> </a> <ul id="toc-Systematic_encoding_procedure-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> 4 Properties </div> </a> <button aria-controls="toc-Properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Properties subsection </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Remarks" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Remarks"> <div class="vector-toc-text"> 4.1 Remarks </div> </a> <ul id="toc-Remarks-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-BCH_view_decoders" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#BCH_view_decoders"> <div class="vector-toc-text"> 5 BCH view decoders </div> </a> <button aria-controls="toc-BCH_view_decoders-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle BCH view decoders subsection </button> <ul id="toc-BCH_view_decoders-sublist" class="vector-toc-list"> <li id="toc-Peterson–Gorenstein–Zierler_decoder" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Peterson–Gorenstein–Zierler_decoder"> <div class="vector-toc-text"> 5.1 Peterson–Gorenstein–Zierler decoder </div> </a> <ul id="toc-Peterson–Gorenstein–Zierler_decoder-sublist" class="vector-toc-list"> <li id="toc-Formulation" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Formulation"> <div class="vector-toc-text"> 5.1.1 Formulation </div> </a> <ul id="toc-Formulation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Syndrome_decoding" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Syndrome_decoding"> <div class="vector-toc-text"> 5.1.2 Syndrome decoding </div> </a> <ul id="toc-Syndrome_decoding-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Error_locators_and_error_values" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Error_locators_and_error_values"> <div class="vector-toc-text"> 5.1.3 Error locators and error values </div> </a> <ul id="toc-Error_locators_and_error_values-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Error_locator_polynomial" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Error_locator_polynomial"> <div class="vector-toc-text"> 5.1.4 Error locator polynomial </div> </a> <ul id="toc-Error_locator_polynomial-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Find_the_roots_of_the_error_locator_polynomial" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Find_the_roots_of_the_error_locator_polynomial"> <div class="vector-toc-text"> 5.1.5 Find the roots of the error locator polynomial </div> </a> <ul id="toc-Find_the_roots_of_the_error_locator_polynomial-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Calculate_the_error_values" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Calculate_the_error_values"> <div class="vector-toc-text"> 5.1.6 Calculate the error values </div> </a> <ul id="toc-Calculate_the_error_values-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Calculate_the_error_locations" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Calculate_the_error_locations"> <div class="vector-toc-text"> 5.1.7 Calculate the error locations </div> </a> <ul id="toc-Calculate_the_error_locations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fix_the_errors" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Fix_the_errors"> <div class="vector-toc-text"> 5.1.8 Fix the errors </div> </a> <ul id="toc-Fix_the_errors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Example"> <div class="vector-toc-text"> 5.1.9 Example </div> </a> <ul id="toc-Example-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Berlekamp–Massey_decoder" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Berlekamp–Massey_decoder"> <div class="vector-toc-text"> 5.2 Berlekamp–Massey decoder </div> </a> <ul id="toc-Berlekamp–Massey_decoder-sublist" class="vector-toc-list"> <li id="toc-Example_2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Example_2"> <div class="vector-toc-text"> 5.2.1 Example </div> </a> <ul id="toc-Example_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Euclidean_decoder" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Euclidean_decoder"> <div class="vector-toc-text"> 5.3 Euclidean decoder </div> </a> <ul id="toc-Euclidean_decoder-sublist" class="vector-toc-list"> <li id="toc-Example_3" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Example_3"> <div class="vector-toc-text"> 5.3.1 Example </div> </a> <ul id="toc-Example_3-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Decoder_using_discrete_Fourier_transform" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Decoder_using_discrete_Fourier_transform"> <div class="vector-toc-text"> 5.4 Decoder using discrete Fourier transform </div> </a> <ul id="toc-Decoder_using_discrete_Fourier_transform-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Decoding_beyond_the_error-correction_bound" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Decoding_beyond_the_error-correction_bound"> <div class="vector-toc-text"> 5.5 Decoding beyond the error-correction bound </div> </a> <ul id="toc-Decoding_beyond_the_error-correction_bound-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Soft-decoding" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Soft-decoding"> <div class="vector-toc-text"> 5.6 Soft-decoding </div> </a> <ul id="toc-Soft-decoding-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-MATLAB_example" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#MATLAB_example"> <div class="vector-toc-text"> 5.7 MATLAB example </div> </a> <ul id="toc-MATLAB_example-sublist" class="vector-toc-list"> <li id="toc-Encoder" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Encoder"> <div class="vector-toc-text"> 5.7.1 Encoder </div> </a> <ul id="toc-Encoder-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Decoder" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Decoder"> <div class="vector-toc-text"> 5.7.2 Decoder </div> </a> <ul id="toc-Decoder-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Reed_Solomon_original_view_decoders" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Reed_Solomon_original_view_decoders"> <div class="vector-toc-text"> 6 Reed Solomon original view decoders </div> </a> <button aria-controls="toc-Reed_Solomon_original_view_decoders-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Reed Solomon original view decoders subsection </button> <ul id="toc-Reed_Solomon_original_view_decoders-sublist" class="vector-toc-list"> <li id="toc-Theoretical_decoder" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Theoretical_decoder"> <div class="vector-toc-text"> 6.1 Theoretical decoder </div> </a> <ul id="toc-Theoretical_decoder-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Berlekamp_Welch_decoder" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Berlekamp_Welch_decoder"> <div class="vector-toc-text"> 6.2 Berlekamp Welch decoder </div> </a> <ul id="toc-Berlekamp_Welch_decoder-sublist" class="vector-toc-list"> <li id="toc-Example_4" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Example_4"> <div class="vector-toc-text"> 6.2.1 Example </div> </a> <ul id="toc-Example_4-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Gao_decoder" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Gao_decoder"> <div class="vector-toc-text"> 6.3 Gao decoder </div> </a> <ul id="toc-Gao_decoder-sublist" class="vector-toc-list"> <li id="toc-Example_5" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Example_5"> <div class="vector-toc-text"> 6.3.1 Example </div> </a> <ul id="toc-Example_5-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 7 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 8 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 9 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 10 External links </div> </a> <button aria-controls="toc-External_links-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle External links subsection </button> <ul id="toc-External_links-sublist" class="vector-toc-list"> <li id="toc-Information_and_tutorials" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Information_and_tutorials"> <div class="vector-toc-text"> 10.1 Information and tutorials </div> </a> <ul id="toc-Information_and_tutorials-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Implementations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Implementations"> <div class="vector-toc-text"> 10.2 Implementations </div> </a> <ul id="toc-Implementations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Reed–Solomon error correction</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 21 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-21" aria-hidden="true" > 21 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Reed-Solomon" title="Reed-Solomon – Catalan" lang="ca" hreflang="ca" data-title="Reed-Solomon" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Reedovy%E2%80%93Solomonovy_k%C3%B3dy" title="Reedovy–Solomonovy kódy – Czech" lang="cs" hreflang="cs" data-title="Reedovy–Solomonovy kódy" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Reed-Solomon-fejlkorrektion" title="Reed-Solomon-fejlkorrektion – Danish" lang="da" hreflang="da" data-title="Reed-Solomon-fejlkorrektion" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Reed-Solomon-Code" title="Reed-Solomon-Code – German" lang="de" hreflang="de" data-title="Reed-Solomon-Code" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Reed-Solomon" title="Reed-Solomon – Spanish" lang="es" hreflang="es" data-title="Reed-Solomon" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%B5%D8%AD%DB%8C%D8%AD_%D8%AE%D8%B7%D8%A7%DB%8C_%D8%B1%DB%8C%D8%AF-%D8%B3%D8%A7%D9%84%D8%A7%D9%85%D9%88%D9%86" title="تصحیح خطای رید-سالامون – Persian" lang="fa" hreflang="fa" data-title="تصحیح خطای رید-سالامون" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Code_de_Reed-Solomon" title="Code de Reed-Solomon – French" lang="fr" hreflang="fr" data-title="Code de Reed-Solomon" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%A6%AC%EB%93%9C_%EC%86%94%EB%A1%9C%EB%AA%AC_%EB%B6%80%ED%98%B8" title="리드 솔로몬 부호 – Korean" lang="ko" hreflang="ko" data-title="리드 솔로몬 부호" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Codice_Reed-Solomon" title="Codice Reed-Solomon – Italian" lang="it" hreflang="it" data-title="Codice Reed-Solomon" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A7%D7%95%D7%93_%D7%A8%D7%99%D7%93-%D7%A1%D7%95%D7%9C%D7%95%D7%9E%D7%95%D7%9F" title="קוד ריד-סולומון – Hebrew" lang="he" hreflang="he" data-title="קוד ריד-סולומון" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A0%D0%B8%D0%B9%D0%B4-%D0%A1%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D0%BD%D1%8B_%D0%BA%D0%BE%D0%B4" title="Рийд-Соломоны код – Mongolian" lang="mn" hreflang="mn" data-title="Рийд-Соломоны код" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target">Монгол</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Reed-Solomoncode" title="Reed-Solomoncode – Dutch" lang="nl" hreflang="nl" data-title="Reed-Solomoncode" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%AA%E3%83%BC%E3%83%89%E3%83%BB%E3%82%BD%E3%83%AD%E3%83%A2%E3%83%B3%E7%AC%A6%E5%8F%B7" title="リード・ソロモン符号 – Japanese" lang="ja" hreflang="ja" data-title="リード・ソロモン符号" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Kodowanie_korekcyjne_Reeda-Solomona" title="Kodowanie korekcyjne Reeda-Solomona – Polish" lang="pl" hreflang="pl" data-title="Kodowanie korekcyjne Reeda-Solomona" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/C%C3%B3digo_Reed%E2%80%93Solomon" title="Código Reed–Solomon – Portuguese" lang="pt" hreflang="pt" data-title="Código Reed–Solomon" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%BE%D0%B4_%D0%A0%D0%B8%D0%B4%D0%B0_%E2%80%94_%D0%A1%D0%BE%D0%BB%D0%BE%D0%BC%D0%BE%D0%BD%D0%B0" title="Код Рида — Соломона – Russian" lang="ru" hreflang="ru" data-title="Код Рида — Соломона" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Reed%E2%80%93Solomon_error_correction" title="Reed–Solomon error correction – Simple English" lang="en-simple" hreflang="en-simple" data-title="Reed–Solomon error correction" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Reed%E2%80%93Solomon-koodit" title="Reed–Solomon-koodit – Finnish" lang="fi" hreflang="fi" data-title="Reed–Solomon-koodit" 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data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Error-correcting codes</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above">Reed–Solomon codes</th></tr><tr><th scope="row" class="infobox-label">Named after</th><td class="infobox-data"><a href="/wiki/Irving_S._Reed" title="Irving S. Reed">Irving S. Reed</a> and <a href="/wiki/Gustave_Solomon" title="Gustave Solomon">Gustave Solomon</a></td></tr><tr><th colspan="2" class="infobox-header" style="background:#ccf;">Classification</th></tr><tr><th scope="row" class="infobox-label">Hierarchy</th><td class="infobox-data"><a href="/wiki/Linear_block_code" class="mw-redirect" title="Linear block code">Linear block code</a> <a href="/wiki/Polynomial_code" title="Polynomial code">Polynomial code</a> Reed–Solomon code</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Block_code#The_block_length_n" title="Block code">Block length</a></th><td class="infobox-data">n</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Block_code#The_message_length_k" title="Block code">Message length</a></th><td class="infobox-data">k</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Block_code#The_distance_d" title="Block code">Distance</a></th><td class="infobox-data">n − k + 1</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Block_code#The_alphabet_Σ" title="Block code">Alphabet size</a></th><td class="infobox-data">q = pm ≥ n  (p prime) Often n = q − 1.</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Block_code#Popular_notation" title="Block code">Notation</a></th><td class="infobox-data">[n, k, n − k + 1]q-code</td></tr><tr><th colspan="2" class="infobox-header" style="background:#ccf;">Algorithms</th></tr><tr><td colspan="2" class="infobox-full-data"><a href="/wiki/Berlekamp%E2%80%93Massey_algorithm" title="Berlekamp–Massey algorithm">Berlekamp–Massey</a> <a href="/wiki/Euclidean_algorithm" title="Euclidean algorithm">Euclidean</a> et al.</td></tr><tr><th colspan="2" class="infobox-header" style="background:#ccf;">Properties</th></tr><tr><td colspan="2" class="infobox-full-data"><a href="/wiki/Maximum_distance_separable_code" class="mw-redirect" title="Maximum distance separable code">Maximum-distance separable code</a></td></tr><tr><td colspan="2" class="infobox-navbar"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Infobox_code" title="Template:Infobox code"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Infobox_code" title="Template talk:Infobox code"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Infobox_code" title="Special:EditPage/Template:Infobox code"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> In <a href="/wiki/Information_theory" title="Information theory">information theory</a> and <a href="/wiki/Coding_theory" title="Coding theory">coding theory</a>, Reed–Solomon codes are a group of <a href="/wiki/Error-correcting_code" class="mw-redirect" title="Error-correcting code">error-correcting codes</a> that were introduced by <a href="/wiki/Irving_S._Reed" title="Irving S. Reed">Irving S. Reed</a> and <a href="/wiki/Gustave_Solomon" title="Gustave Solomon">Gustave Solomon</a> in 1960.<a href="#cite_note-ReedSolomon-1">[1]</a> They have many applications, including consumer technologies such as <a href="/wiki/MiniDisc" title="MiniDisc">MiniDiscs</a>, <a href="/wiki/CD" class="mw-redirect" title="CD">CDs</a>, <a href="/wiki/DVD" title="DVD">DVDs</a>, <a href="/wiki/Blu-ray" title="Blu-ray">Blu-ray</a> discs, <a href="/wiki/QR_code" title="QR code">QR codes</a>, <a href="/wiki/Data_Matrix" title="Data Matrix">Data Matrix</a>, <a href="/wiki/Data_transmission" class="mw-redirect" title="Data transmission">data transmission</a> technologies such as <a href="/wiki/DSL" class="mw-redirect" title="DSL">DSL</a> and <a href="/wiki/WiMAX" title="WiMAX">WiMAX</a>, <a href="/wiki/Broadcasting" title="Broadcasting">broadcast</a> systems such as satellite communications, <a href="/wiki/Digital_Video_Broadcasting" class="mw-redirect" title="Digital Video Broadcasting">DVB</a> and <a href="/wiki/ATSC_Standards" class="mw-redirect" title="ATSC Standards">ATSC</a>, and storage systems such as <a href="/wiki/RAID_6" class="mw-redirect" title="RAID 6">RAID 6</a>. Reed–Solomon codes operate on a block of data treated as a set of <a href="/wiki/Finite_field" title="Finite field">finite-field</a> elements called symbols. Reed–Solomon codes are able to detect and correct multiple symbol errors. By adding t = n − k check symbols to the data, a Reed–Solomon code can detect (but not correct) any combination of up to t erroneous symbols, or locate and correct up to ⌊t/2⌋ erroneous symbols at unknown locations. As an <a href="/wiki/Erasure_code" title="Erasure code">erasure code</a>, it can correct up to t erasures at locations that are known and provided to the algorithm, or it can detect and correct combinations of errors and erasures. Reed–Solomon codes are also suitable as multiple-<a href="/wiki/Burst_error" title="Burst error">burst</a> bit-error correcting codes, since a sequence of b + 1 consecutive bit errors can affect at most two symbols of size b. The choice of t is up to the designer of the code and may be selected within wide limits. There are two basic types of Reed–Solomon codes – original view and <a href="/wiki/BCH_code" title="BCH code">BCH</a> view – with BCH view being the most common, as BCH view decoders are faster and require less working storage than original view decoders. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=1" title="Edit section: History">edit</a>]</div> Reed–Solomon codes were developed in 1960 by <a href="/wiki/Irving_S._Reed" title="Irving S. Reed">Irving S. Reed</a> and <a href="/wiki/Gustave_Solomon" title="Gustave Solomon">Gustave Solomon</a>, who were then staff members of <a href="/wiki/MIT_Lincoln_Laboratory" title="MIT Lincoln Laboratory">MIT Lincoln Laboratory</a>. Their seminal article was titled "Polynomial Codes over Certain Finite Fields".<a href="#cite_note-ReedSolomon-1">[1]</a> The original encoding scheme described in the Reed and Solomon article used a variable polynomial based on the message to be encoded where only a fixed set of values (evaluation points) to be encoded are known to encoder and decoder. The original theoretical decoder generated potential polynomials based on subsets of k (unencoded message length) out of n (encoded message length) values of a received message, choosing the most popular polynomial as the correct one, which was impractical for all but the simplest of cases. This was initially resolved by changing the original scheme to a <a href="/wiki/BCH_code" title="BCH code">BCH-code</a>-like scheme based on a fixed polynomial known to both encoder and decoder, but later, practical decoders based on the original scheme were developed, although slower than the BCH schemes. The result of this is that there are two main types of Reed–Solomon codes: ones that use the original encoding scheme and ones that use the BCH encoding scheme. Also in 1960, a practical fixed polynomial decoder for <a href="/wiki/BCH_codes" class="mw-redirect" title="BCH codes">BCH codes</a> developed by <a href="/wiki/Daniel_Gorenstein" title="Daniel Gorenstein">Daniel Gorenstein</a> and Neal Zierler was described in an <a href="/wiki/MIT_Lincoln_Laboratory" title="MIT Lincoln Laboratory">MIT Lincoln Laboratory</a> report by Zierler in January 1960 and later in an article in June 1961.<a href="#cite_note-2">[2]</a> The Gorenstein–Zierler decoder and the related work on BCH codes are described in a book "Error-Correcting Codes" by <a href="/wiki/W._Wesley_Peterson" title="W. Wesley Peterson">W. Wesley Peterson</a> (1961).<a href="#cite_note-Peterson61-3">[3]</a>[<a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">page needed</a>] By 1963 (or possibly earlier), J.J. Stone (and others)[<a href="/wiki/Wikipedia:Manual_of_Style/Words_to_watch#Unsupported_attributions" title="Wikipedia:Manual of Style/Words to watch">who?</a>] recognized that Reed–Solomon codes could use the BCH scheme of using a fixed generator polynomial, making such codes a special class of BCH codes,<a href="#cite_note-Peterson96-4">[4]</a> but Reed–Solomon codes based on the original encoding scheme are not a class of BCH codes, and depending on the set of evaluation points, they are not even <a href="/wiki/Cyclic_code" title="Cyclic code">cyclic codes</a>. In 1969, an improved BCH scheme decoder was developed by <a href="/wiki/Elwyn_Berlekamp" title="Elwyn Berlekamp">Elwyn Berlekamp</a> and <a href="/wiki/James_Massey" title="James Massey">James Massey</a> and has since been known as the <a href="/wiki/Berlekamp%E2%80%93Massey_algorithm" title="Berlekamp–Massey algorithm">Berlekamp–Massey decoding algorithm</a>. In 1975, another improved BCH scheme decoder was developed by Yasuo Sugiyama, based on the <a href="/wiki/Extended_Euclidean_algorithm" title="Extended Euclidean algorithm">extended Euclidean algorithm</a>.<a href="#cite_note-5">[5]</a> <figure typeof="mw:File/Thumb"><a href="/wiki/File:DVB-vs-DVB-X2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/DVB-vs-DVB-X2.png/500px-DVB-vs-DVB-X2.png" decoding="async" width="350" height="196" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/DVB-vs-DVB-X2.png/960px-DVB-vs-DVB-X2.png 1.5x" data-file-width="1969" data-file-height="1101" /></a><figcaption></figcaption></figure> In 1977, Reed–Solomon codes were implemented in the <a href="/wiki/Voyager_program" title="Voyager program">Voyager program</a> in the form of <a href="/wiki/Concatenated_error_correction_code" title="Concatenated error correction code">concatenated error correction codes</a>. The first commercial application in mass-produced consumer products appeared in 1982 with the <a href="/wiki/Compact_disc" title="Compact disc">compact disc</a>, where two <a href="/wiki/Forward_error_correction#Interleaving" class="mw-redirect" title="Forward error correction">interleaved</a> Reed–Solomon codes are used. Today, Reed–Solomon codes are widely implemented in <a href="/wiki/Data_storage_device" class="mw-redirect" title="Data storage device">digital storage</a> devices and <a href="/wiki/Data_transmission" class="mw-redirect" title="Data transmission">digital communication</a> standards, though they are being slowly replaced by <a href="/wiki/BCH_code" title="BCH code">Bose–Chaudhuri–Hocquenghem (BCH) codes</a>. For example, Reed–Solomon codes are used in the <a href="/wiki/Digital_Video_Broadcasting" class="mw-redirect" title="Digital Video Broadcasting">Digital Video Broadcasting</a> (DVB) standard <a href="/wiki/DVB-S" title="DVB-S">DVB-S</a>, in conjunction with a <a href="/wiki/Convolutional_coding" class="mw-redirect" title="Convolutional coding">convolutional</a> <a href="/wiki/Inner_code" class="mw-redirect" title="Inner code">inner code</a>, but BCH codes are used with <a href="/wiki/Low-density_parity-check_code" title="Low-density parity-check code">LDPC</a> in its successor, <a href="/wiki/DVB-S2" title="DVB-S2">DVB-S2</a>. In 1986, an original scheme decoder known as the <a href="/wiki/Berlekamp%E2%80%93Welch_algorithm" title="Berlekamp–Welch algorithm">Berlekamp–Welch algorithm</a> was developed. In 1996, variations of original scheme decoders called list decoders or soft decoders were developed by Madhu Sudan and others, and work continues on these types of decoders (see <a href="/wiki/Guruswami%E2%80%93Sudan_list_decoding_algorithm" title="Guruswami–Sudan list decoding algorithm">Guruswami–Sudan list decoding algorithm</a>). In 2002, another original scheme decoder was developed by Shuhong Gao, based on the <a href="/wiki/Extended_Euclidean_algorithm" title="Extended Euclidean algorithm">extended Euclidean algorithm</a>.<a href="#cite_note-gao-6">[6]</a> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=2" title="Edit section: Applications">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Data_storage">Data storage</h3>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=3" title="Edit section: Data storage">edit</a>]</div> Reed–Solomon coding is very widely used in mass storage systems to correct the burst errors associated with media defects. Reed–Solomon coding is a key component of the <a href="/wiki/Compact_disc" title="Compact disc">compact disc</a>. It was the first use of strong error correction coding in a mass-produced consumer product, and <a href="/wiki/Digital_audio_tape" class="mw-redirect" title="Digital audio tape">DAT</a> and <a href="/wiki/DVD" title="DVD">DVD</a> use similar schemes. In the CD, two layers of Reed–Solomon coding separated by a 28-way <a href="/wiki/Convolution" title="Convolution">convolutional</a> <a href="/wiki/Interleaver" class="mw-redirect" title="Interleaver">interleaver</a> yields a scheme called Cross-Interleaved Reed–Solomon Coding (<a href="/wiki/Cross-interleaved_Reed%E2%80%93Solomon_coding" title="Cross-interleaved Reed–Solomon coding">CIRC</a>). The first element of a CIRC decoder is a relatively weak inner (32,28) Reed–Solomon code, shortened from a (255,251) code with 8-bit symbols. This code can correct up to 2 byte errors per 32-byte block. More importantly, it flags as erasures any uncorrectable blocks, i.e., blocks with more than 2 byte errors. The decoded 28-byte blocks, with erasure indications, are then spread by the deinterleaver to different blocks of the (28,24) outer code. Thanks to the deinterleaving, an erased 28-byte block from the inner code becomes a single erased byte in each of 28 outer code blocks. The outer code easily corrects this, since it can handle up to 4 such erasures per block. The result is a CIRC that can completely correct error bursts up to 4000 bits, or about 2.5 mm on the disc surface. This code is so strong that most CD playback errors are almost certainly caused by tracking errors that cause the laser to jump track, not by uncorrectable error bursts.<a href="#cite_note-7">[7]</a> DVDs use a similar scheme, but with much larger blocks, a (208,192) inner code, and a (182,172) outer code. Reed–Solomon error correction is also used in <a href="/wiki/Parchive" title="Parchive">parchive</a> files which are commonly posted accompanying multimedia files on <a href="/wiki/USENET" class="mw-redirect" title="USENET">USENET</a>. The distributed online storage service <a href="/wiki/Wuala" title="Wuala">Wuala</a> (discontinued in 2015) also used Reed–Solomon when breaking up files. <div class="mw-heading mw-heading3"><h3 id="Bar_code">Bar code</h3>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=4" title="Edit section: Bar code">edit</a>]</div> Almost all two-dimensional bar codes such as <a href="/wiki/PDF-417" class="mw-redirect" title="PDF-417">PDF-417</a>, <a href="/wiki/MaxiCode" title="MaxiCode">MaxiCode</a>, <a href="/wiki/Datamatrix" class="mw-redirect" title="Datamatrix">Datamatrix</a>, <a href="/wiki/QR_Code" class="mw-redirect" title="QR Code">QR Code</a>, <a href="/wiki/Aztec_Code" title="Aztec Code">Aztec Code</a> and <a href="/wiki/Han_Xin_code" title="Han Xin code">Han Xin code</a> use Reed–Solomon error correction to allow correct reading even if a portion of the bar code is damaged. When the bar code scanner cannot recognize a bar code symbol, it will treat it as an erasure. Reed–Solomon coding is less common in one-dimensional bar codes, but is used by the <a href="/wiki/PostBar" title="PostBar">PostBar</a> symbology. <div class="mw-heading mw-heading3"><h3 id="Data_transmission">Data transmission</h3>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=5" title="Edit section: Data transmission">edit</a>]</div> Specialized forms of Reed–Solomon codes, specifically <a href="/wiki/Cauchy_matrix" title="Cauchy matrix">Cauchy</a>-RS and <a href="/wiki/Vandermonde_matrix" title="Vandermonde matrix">Vandermonde</a>-RS, can be used to overcome the unreliable nature of data transmission over <a href="/wiki/Binary_erasure_channel" title="Binary erasure channel">erasure channels</a>. The encoding process assumes a code of RS(N, K) which results in N codewords of length N symbols each storing K symbols of data, being generated, that are then sent over an erasure channel. Any combination of K codewords received at the other end is enough to reconstruct all of the N codewords. The code rate is generally set to 1/2 unless the channel's erasure likelihood can be adequately modelled and is seen to be less. In conclusion, N is usually 2K, meaning that at least half of all the codewords sent must be received in order to reconstruct all of the codewords sent. Reed–Solomon codes are also used in <a href="/wiki/XDSL" class="mw-redirect" title="XDSL">xDSL</a> systems and <a href="/wiki/CCSDS" class="mw-redirect" title="CCSDS">CCSDS</a>'s <a href="/wiki/Space_Communications_Protocol_Specifications" title="Space Communications Protocol Specifications">Space Communications Protocol Specifications</a> as a form of <a href="/wiki/Forward_error_correction" class="mw-redirect" title="Forward error correction">forward error correction</a>. <div class="mw-heading mw-heading3"><h3 id="Space_transmission">Space transmission</h3>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=6" title="Edit section: Space transmission">edit</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:DeepSpaceFEC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/DeepSpaceFEC.png/500px-DeepSpaceFEC.png" decoding="async" width="350" height="199" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/DeepSpaceFEC.png/960px-DeepSpaceFEC.png 1.5x" data-file-width="1276" data-file-height="724" /></a><figcaption> Deep-space concatenated coding system.<a href="#cite_note-8">[8]</a> Notation: RS(255, 223) + <a href="/wiki/Convolutional_codes" class="mw-redirect" title="Convolutional codes">CC</a> ("constraint length" = 7, code rate = 1/2).</figcaption></figure> One significant application of Reed–Solomon coding was to encode the digital pictures sent back by the <a href="/wiki/Voyager_program" title="Voyager program">Voyager program</a>. Voyager introduced Reed–Solomon coding <a href="/wiki/Concatenated_code" class="mw-redirect" title="Concatenated code">concatenated</a> with <a href="/wiki/Convolutional_code" title="Convolutional code">convolutional codes</a>, a practice that has since become very widespread in deep space and satellite (e.g., direct digital broadcasting) communications. <a href="/wiki/Viterbi_decoder" title="Viterbi decoder">Viterbi decoders</a> tend to produce errors in short bursts. Correcting these burst errors is a job best done by short or simplified Reed–Solomon codes. Modern versions of concatenated Reed–Solomon/Viterbi-decoded convolutional coding were and are used on the <a href="/wiki/Mars_Pathfinder" title="Mars Pathfinder">Mars Pathfinder</a>, <a href="/wiki/Galileo_probe" class="mw-redirect" title="Galileo probe">Galileo</a>, <a href="/wiki/Mars_Exploration_Rover" title="Mars Exploration Rover">Mars Exploration Rover</a> and <a href="/wiki/Cassini_probe" class="mw-redirect" title="Cassini probe">Cassini</a> missions, where they perform within about 1–1.5 <a href="/wiki/Decibel" title="Decibel">dB</a> of the ultimate limit, the <a href="/wiki/Channel_capacity" title="Channel capacity">Shannon capacity</a>. These concatenated codes are now being replaced by more powerful <a href="/wiki/Turbo_code" title="Turbo code">turbo codes</a>: <table class="wikitable"> <caption>Channel coding schemes used by NASA missions<a href="#cite_note-Andrews,_2007-9">[9]</a> </caption> <tbody><tr> <th>Years</th> <th>Code</th> <th>Mission(s) </th></tr> <tr> <td>1958–present</td> <td>Uncoded</td> <td>Explorer, Mariner, many others </td></tr> <tr> <td>1968–1978</td> <td><a href="/wiki/Convolutional_codes" class="mw-redirect" title="Convolutional codes">convolutional codes</a> (CC) (25, 1/2)</td> <td>Pioneer, Venus </td></tr> <tr> <td>1969–1975</td> <td><a href="/wiki/Reed-Muller_code" class="mw-redirect" title="Reed-Muller code">Reed-Muller code</a> (32, 6)</td> <td>Mariner, Viking </td></tr> <tr> <td>1977–present</td> <td><a href="/wiki/Binary_Golay_code" title="Binary Golay code">Binary Golay code</a></td> <td>Voyager </td></tr> <tr> <td>1977–present</td> <td>RS(255, 223) + CC(7, 1/2)</td> <td>Voyager, Galileo, many others </td></tr> <tr> <td>1989–2003</td> <td>RS(255, 223) + CC(7, 1/3)</td> <td>Voyager </td></tr> <tr> <td>1989–2003</td> <td>RS(255, 223) + CC(14, 1/4)</td> <td>Galileo </td></tr> <tr> <td>1996–present</td> <td>RS + CC (15, 1/6)</td> <td>Cassini, Mars Pathfinder, others </td></tr> <tr> <td>2004–present</td> <td><a href="/wiki/Turbo_codes" class="mw-redirect" title="Turbo codes">Turbo codes</a><a href="#cite_note-10">[nb 1]</a> </td> <td>Messenger, Stereo, MRO, others </td></tr> <tr> <td>est. 2009</td> <td><a href="/wiki/LDPC_codes" class="mw-redirect" title="LDPC codes">LDPC codes</a></td> <td>Constellation, MSL </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Constructions_(encoding)">Constructions (encoding)</h2>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=7" title="Edit section: Constructions (encoding)">edit</a>]</div> The Reed–Solomon code is actually a family of codes, where every code is characterised by three parameters: an <a href="/wiki/Block_code#The_alphabet_.CE.A3" title="Block code">alphabet</a> size q, a <a href="/wiki/Block_code#The_block_length_n" title="Block code">block length</a> n, and a <a href="/wiki/Block_code#The_message_length_k" title="Block code">message length</a> k, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k<n\leq q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo><</mo> <mi>n</mi> <mo>≤</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k<n\leq q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78b4f2248f86608e267c8d6e9bf9f38beb064fc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.872ex; height:2.509ex;" alt="{\displaystyle k<n\leq q}" />. The set of alphabet symbols is interpreted as the <a href="/wiki/Finite_field" title="Finite field">finite field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" /> of order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" />, and thus, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" /> must be a <a href="/wiki/Prime_power" title="Prime power">prime power</a>. In the most useful parameterizations of the Reed–Solomon code, the block length is usually some constant multiple of the message length, that is, the <a href="/wiki/Block_code#The_rate_R" title="Block code">rate</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R={\frac {k}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>n</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R={\frac {k}{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8fb6f00490d6179856cc45e37f1bd60677f7ecf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.093ex; height:5.343ex;" alt="{\displaystyle R={\frac {k}{n}}}" /> is some constant, and furthermore, the block length is either equal to the alphabet size or one less than it, i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b88eedc2745402e0d99d3886e05ea683816aecf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.563ex; height:2.009ex;" alt="{\displaystyle n=q}" /> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=q-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mi>q</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=q-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1989c9f290cfc2e56144b9fe0790657d730f0ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.566ex; height:2.509ex;" alt="{\displaystyle n=q-1}" />.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <div class="mw-heading mw-heading3"><h3 id="Reed_&_Solomon's_original_view:_The_codeword_as_a_sequence_of_values">Reed & Solomon's original view: The codeword as a sequence of values</h3>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=8" title="Edit section: Reed & Solomon's original view: The codeword as a sequence of values">edit</a>]</div> There are different encoding procedures for the Reed–Solomon code, and thus, there are different ways to describe the set of all codewords. In the original view of Reed and Solomon, every codeword of the Reed–Solomon code is a sequence of function values of a polynomial of degree less than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" />.<a href="#cite_note-ReedSolomon-1">[1]</a> In order to obtain a codeword of the Reed–Solomon code, the message symbols (each within the q-sized alphabet) are treated as the coefficients of a polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" /> of degree less than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" />, over the finite field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" /> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" /> elements. In turn, the polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" /> is evaluated at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\leq q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≤</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\leq q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a2267f3d1b2d1d38418583aa96e623c684ad359" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.563ex; height:2.343ex;" alt="{\displaystyle n\leq q}" /> distinct points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},\dots ,a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},\dots ,a_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/852381be25b656d697c7a4a9634d3dc4c182d833" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.911ex; height:2.009ex;" alt="{\displaystyle a_{1},\dots ,a_{n}}" /> of the field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" />, and the sequence of values is the corresponding codeword. Common choices for a set of evaluation points include <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0,1,2,\dots ,n-1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0,1,2,\dots ,n-1\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a274c78335c05b113ff9a19b66526a9da869fbfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.456ex; height:2.843ex;" alt="{\displaystyle \{0,1,2,\dots ,n-1\}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0,1,\alpha ,\alpha ^{2},\dots ,\alpha ^{n-2}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mi>α</mi> <mo>,</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0,1,\alpha ,\alpha ^{2},\dots ,\alpha ^{n-2}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb99dd9adcb2aa9fc800c86b8be528af7c2a657a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.766ex; height:3.176ex;" alt="{\displaystyle \{0,1,\alpha ,\alpha ^{2},\dots ,\alpha ^{n-2}\}}" />, or for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n<q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo><</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n<q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/145a4cb4ddb9a99b2a6263ed1420f60d130f7d35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.563ex; height:2.176ex;" alt="{\displaystyle n<q}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,\alpha ,\alpha ^{2},\dots ,\alpha ^{n-1}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mi>α</mi> <mo>,</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,\alpha ,\alpha ^{2},\dots ,\alpha ^{n-1}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f9ba4dc7845fca2e4dcf9711e11994e1ae019ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.57ex; height:3.176ex;" alt="{\displaystyle \{1,\alpha ,\alpha ^{2},\dots ,\alpha ^{n-1}\}}" />, ... , where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /> is a <a href="/wiki/Primitive_element_(finite_field)" title="Primitive element (finite field)">primitive element</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" />. Formally, the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11de80478fce9090e43eed19100b37cc841661e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\displaystyle \mathbf {C} }" /> of codewords of the Reed–Solomon code is defined as follows: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} ={\Bigl \{}\;{\bigl (}p(a_{1}),p(a_{2}),\dots ,p(a_{n}){\bigr )}\;{\Big |}\;p{\text{ is a polynomial over }}F{\text{ of degree }}<k\;{\Bigr \}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">{</mo> </mrow> </mrow> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>p</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mi>p</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>p</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">|</mo> </mrow> </mrow> <mspace width="thickmathspace"></mspace> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is a polynomial over </mtext> </mrow> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> of degree </mtext> </mrow> <mo><</mo> <mi>k</mi> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">}</mo> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {C} ={\Bigl \{}\;{\bigl (}p(a_{1}),p(a_{2}),\dots ,p(a_{n}){\bigr )}\;{\Big |}\;p{\text{ is a polynomial over }}F{\text{ of degree }}<k\;{\Bigr \}}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e97b14d102072e8872be43fd3bfd4d25fdd33dd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:75.125ex; height:4.843ex;" alt="{\displaystyle \mathbf {C} ={\Bigl \{}\;{\bigl (}p(a_{1}),p(a_{2}),\dots ,p(a_{n}){\bigr )}\;{\Big |}\;p{\text{ is a polynomial over }}F{\text{ of degree }}<k\;{\Bigr \}}\,.}" /> Since any two distinct polynomials of degree less than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" /> agree in at most <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21363ebd7038c93aae93127e7d910fc1b2e2c745" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.214ex; height:2.343ex;" alt="{\displaystyle k-1}" /> points, this means that any two codewords of the Reed–Solomon code disagree in at least <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-(k-1)=n-k+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-(k-1)=n-k+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bce2721e0474c79220d2f7a13884fa48998ea3cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.806ex; height:2.843ex;" alt="{\displaystyle n-(k-1)=n-k+1}" /> positions. Furthermore, there are two polynomials that do agree in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21363ebd7038c93aae93127e7d910fc1b2e2c745" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.214ex; height:2.343ex;" alt="{\displaystyle k-1}" /> points but are not equal, and thus, the <a href="/wiki/Block_code#The_distance_d" title="Block code">distance</a> of the Reed–Solomon code is exactly <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d=n-k+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d=n-k+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0990fa9c410b95fed78e55519575a44021a19f19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.764ex; height:2.343ex;" alt="{\displaystyle d=n-k+1}" />. Then the relative distance is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta =d/n=1-k/n+1/n=1-R+1/n\sim 1-R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo>=</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>R</mi> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo>∼</mo> <mn>1</mn> <mo>−</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta =d/n=1-k/n+1/n=1-R+1/n\sim 1-R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96b5b8c599a4e0dc72caf6470946e2d77b6df940" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.641ex; height:2.843ex;" alt="{\displaystyle \delta =d/n=1-k/n+1/n=1-R+1/n\sim 1-R}" />, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=k/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=k/n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/413a3d4d51a7ee9b5b884497ab1dda55a18d0e99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.631ex; height:2.843ex;" alt="{\displaystyle R=k/n}" /> is the rate. This trade-off between the relative distance and the rate is asymptotically optimal since, by the <a href="/wiki/Singleton_bound" title="Singleton bound">Singleton bound</a>, every code satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta +R\leq 1+1/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo>+</mo> <mi>R</mi> <mo>≤</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta +R\leq 1+1/n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1b7f3a86afa445a98c5433e0a86d1a34eb7ea5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.474ex; height:2.843ex;" alt="{\displaystyle \delta +R\leq 1+1/n}" />. Being a code that achieves this optimal trade-off, the Reed–Solomon code belongs to the class of <a href="/wiki/Maximum_distance_separable_code" class="mw-redirect" title="Maximum distance separable code">maximum distance separable codes</a>. While the number of different polynomials of degree less than k and the number of different messages are both equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q^{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc9b1525f5a653e19f9fd37fd2701a768e171632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.168ex; height:3.009ex;" alt="{\displaystyle q^{k}}" />, and thus every message can be uniquely mapped to such a polynomial, there are different ways of doing this encoding. The original construction of Reed & Solomon interprets the message x as the coefficients of the polynomial p, whereas subsequent constructions interpret the message as the values of the polynomial at the first k points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},\dots ,a_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},\dots ,a_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ed5c6512d08d64873d79d51a42e6b057007d1f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.781ex; height:2.009ex;" alt="{\displaystyle a_{1},\dots ,a_{k}}" /> and obtain the polynomial p by interpolating these values with a polynomial of degree less than k. The latter encoding procedure, while being slightly less efficient, has the advantage that it gives rise to a <a href="/wiki/Systematic_code" title="Systematic code">systematic code</a>, that is, the original message is always contained as a subsequence of the codeword.<a href="#cite_note-ReedSolomon-1">[1]</a> <div class="mw-heading mw-heading4"><h4 id="Simple_encoding_procedure:_The_message_as_a_sequence_of_coefficients">Simple encoding procedure: The message as a sequence of coefficients</h4>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=9" title="Edit section: Simple encoding procedure: The message as a sequence of coefficients">edit</a>]</div> In the original construction of Reed and Solomon, the message <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=(m_{0},\dots ,m_{k-1})\in F^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=(m_{0},\dots ,m_{k-1})\in F^{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c266019141a2b79dd9c25d79e6869f674b6f7566" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.195ex; height:3.176ex;" alt="{\displaystyle m=(m_{0},\dots ,m_{k-1})\in F^{k}}" /> is mapped to the polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9fdc3d9c861e438bc344ab483c1e5a896fc002d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.934ex; height:2.009ex;" alt="{\displaystyle p_{m}}" /> with <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{m}(a)=\sum _{i=0}^{k-1}m_{i}a^{i}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{m}(a)=\sum _{i=0}^{k-1}m_{i}a^{i}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c3706dcbcbba5e9f62ef04d68d781950dde1f7c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-left: -0.089ex; width:18.717ex; height:7.343ex;" alt="{\displaystyle p_{m}(a)=\sum _{i=0}^{k-1}m_{i}a^{i}\,.}" /> The codeword of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" /> is obtained by evaluating <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9fdc3d9c861e438bc344ab483c1e5a896fc002d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.934ex; height:2.009ex;" alt="{\displaystyle p_{m}}" /> at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /> different points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0},\dots ,a_{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0},\dots ,a_{n-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af329d1870192111e428e736710c7c169c650285" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.011ex; height:2.009ex;" alt="{\displaystyle a_{0},\dots ,a_{n-1}}" /> of the field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" />.<a href="#cite_note-ReedSolomon-1">[1]</a> Thus the classical encoding function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C:F^{k}\to F^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>:</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C:F^{k}\to F^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0fe838801ac0e5736ce60f7e3c342c268de876c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.254ex; height:2.676ex;" alt="{\displaystyle C:F^{k}\to F^{n}}" /> for the Reed–Solomon code is defined as follows: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(m)={\begin{bmatrix}p_{m}(a_{0})\\p_{m}(a_{1})\\\cdots \\p_{m}(a_{n-1})\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(m)={\begin{bmatrix}p_{m}(a_{0})\\p_{m}(a_{1})\\\cdots \\p_{m}(a_{n-1})\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afa4e3e90d19297cee876396a0015fcce94282f6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:21.769ex; height:12.843ex;" alt="{\displaystyle C(m)={\begin{bmatrix}p_{m}(a_{0})\\p_{m}(a_{1})\\\cdots \\p_{m}(a_{n-1})\end{bmatrix}}}" /> This function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /> is a <a href="/wiki/Linear_mapping" class="mw-redirect" title="Linear mapping">linear mapping</a>, that is, it satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(m)=Am}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(m)=Am}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da15dbc41d01f3e82ce39b503b51800292d081e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.498ex; height:2.843ex;" alt="{\displaystyle C(m)=Am}" /> for the following <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/861eac296135ef99f3b1b36f807616a9d69717f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.446ex; height:2.176ex;" alt="{\displaystyle n\times k}" />-matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /> with elements from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" />: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(m)=Am={\begin{bmatrix}1&a_{0}&a_{0}^{2}&\dots &a_{0}^{k-1}\\1&a_{1}&a_{1}^{2}&\dots &a_{1}^{k-1}\\\vdots &\vdots &\vdots &\ddots &\vdots \\1&a_{n-1}&a_{n-1}^{2}&\dots &a_{n-1}^{k-1}\end{bmatrix}}{\begin{bmatrix}m_{0}\\m_{1}\\\vdots \\m_{k-1}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> <mi>m</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(m)=Am={\begin{bmatrix}1&a_{0}&a_{0}^{2}&\dots &a_{0}^{k-1}\\1&a_{1}&a_{1}^{2}&\dots &a_{1}^{k-1}\\\vdots &\vdots &\vdots &\ddots &\vdots \\1&a_{n-1}&a_{n-1}^{2}&\dots &a_{n-1}^{k-1}\end{bmatrix}}{\begin{bmatrix}m_{0}\\m_{1}\\\vdots \\m_{k-1}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f3aac5d70733eec9747e1f03adffe25d355f979" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.086ex; margin-bottom: -0.252ex; width:55.608ex; height:15.843ex;" alt="{\displaystyle C(m)=Am={\begin{bmatrix}1&a_{0}&a_{0}^{2}&\dots &a_{0}^{k-1}\\1&a_{1}&a_{1}^{2}&\dots &a_{1}^{k-1}\\\vdots &\vdots &\vdots &\ddots &\vdots \\1&a_{n-1}&a_{n-1}^{2}&\dots &a_{n-1}^{k-1}\end{bmatrix}}{\begin{bmatrix}m_{0}\\m_{1}\\\vdots \\m_{k-1}\end{bmatrix}}}" /> This matrix is a <a href="/wiki/Vandermonde_matrix" title="Vandermonde matrix">Vandermonde matrix</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" />. In other words, the Reed–Solomon code is a <a href="/wiki/Linear_code" title="Linear code">linear code</a>, and in the classical encoding procedure, its <a href="/wiki/Linear_code#Properties" title="Linear code">generator matrix</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" />. <div class="mw-heading mw-heading4"><h4 id="Systematic_encoding_procedure:_The_message_as_an_initial_sequence_of_values">Systematic encoding procedure: The message as an initial sequence of values</h4>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=10" title="Edit section: Systematic encoding procedure: The message as an initial sequence of values">edit</a>]</div> There are alternative encoding procedures that produce a <a href="/wiki/Systematic_code" title="Systematic code">systematic</a> Reed–Solomon code. One method uses <a href="/wiki/Lagrange_interpolation" class="mw-redirect" title="Lagrange interpolation">Lagrange interpolation</a> to compute polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9fdc3d9c861e438bc344ab483c1e5a896fc002d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.934ex; height:2.009ex;" alt="{\displaystyle p_{m}}" /> such that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{m}(a_{i})=m_{i}{\text{ for all }}i\in \{0,\dots ,k-1\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mtext> for all </mtext> </mrow> <mi>i</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{m}(a_{i})=m_{i}{\text{ for all }}i\in \{0,\dots ,k-1\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32126b3ed2a8a1e10359eb8d63c7978136556631" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:37.865ex; height:2.843ex;" alt="{\displaystyle p_{m}(a_{i})=m_{i}{\text{ for all }}i\in \{0,\dots ,k-1\}.}" /> Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9fdc3d9c861e438bc344ab483c1e5a896fc002d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.934ex; height:2.009ex;" alt="{\displaystyle p_{m}}" /> is evaluated at the other points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{k},\dots ,a_{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{k},\dots ,a_{n-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59e7379a678a38cf8bccf72f89ef8e1ee357341c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.046ex; height:2.009ex;" alt="{\displaystyle a_{k},\dots ,a_{n-1}}" />. <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(m)={\begin{bmatrix}p_{m}(a_{0})\\p_{m}(a_{1})\\\cdots \\p_{m}(a_{n-1})\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(m)={\begin{bmatrix}p_{m}(a_{0})\\p_{m}(a_{1})\\\cdots \\p_{m}(a_{n-1})\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afa4e3e90d19297cee876396a0015fcce94282f6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:21.769ex; height:12.843ex;" alt="{\displaystyle C(m)={\begin{bmatrix}p_{m}(a_{0})\\p_{m}(a_{1})\\\cdots \\p_{m}(a_{n-1})\end{bmatrix}}}" /> This function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /> is a linear mapping. To generate the corresponding systematic encoding matrix G, multiply the Vandermonde matrix A by the inverse of A's left square submatrix. <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=(A{\text{'s left square submatrix}})^{-1}\cdot A={\begin{bmatrix}1&0&0&\dots &0&g_{1,k+1}&\dots &g_{1,n}\\0&1&0&\dots &0&g_{2,k+1}&\dots &g_{2,n}\\0&0&1&\dots &0&g_{3,k+1}&\dots &g_{3,n}\\\vdots &\vdots &\vdots &&\vdots &\vdots &&\vdots \\0&\dots &0&\dots &1&g_{k,k+1}&\dots &g_{k,n}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>'s left square submatrix</mtext> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>,</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd></mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd></mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=(A{\text{'s left square submatrix}})^{-1}\cdot A={\begin{bmatrix}1&0&0&\dots &0&g_{1,k+1}&\dots &g_{1,n}\\0&1&0&\dots &0&g_{2,k+1}&\dots &g_{2,n}\\0&0&1&\dots &0&g_{3,k+1}&\dots &g_{3,n}\\\vdots &\vdots &\vdots &&\vdots &\vdots &&\vdots \\0&\dots &0&\dots &1&g_{k,k+1}&\dots &g_{k,n}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/339a849ad0936341b27b39e1f03ecbed5cbd4171" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.338ex; width:81.786ex; height:17.843ex;" alt="{\displaystyle G=(A{\text{'s left square submatrix}})^{-1}\cdot A={\begin{bmatrix}1&0&0&\dots &0&g_{1,k+1}&\dots &g_{1,n}\\0&1&0&\dots &0&g_{2,k+1}&\dots &g_{2,n}\\0&0&1&\dots &0&g_{3,k+1}&\dots &g_{3,n}\\\vdots &\vdots &\vdots &&\vdots &\vdots &&\vdots \\0&\dots &0&\dots &1&g_{k,k+1}&\dots &g_{k,n}\end{bmatrix}}}" /> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(m)=Gm}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>G</mi> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(m)=Gm}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff3a4737c628699d66b491532c2f8f6a370b5609" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.582ex; height:2.843ex;" alt="{\displaystyle C(m)=Gm}" /> for the following <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/861eac296135ef99f3b1b36f807616a9d69717f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.446ex; height:2.176ex;" alt="{\displaystyle n\times k}" />-matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" /> with elements from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" />: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(m)=Gm={\begin{bmatrix}1&0&0&\dots &0&g_{1,k+1}&\dots &g_{1,n}\\0&1&0&\dots &0&g_{2,k+1}&\dots &g_{2,n}\\0&0&1&\dots &0&g_{3,k+1}&\dots &g_{3,n}\\\vdots &\vdots &\vdots &&\vdots &\vdots &&\vdots \\0&\dots &0&\dots &1&g_{k,k+1}&\dots &g_{k,n}\end{bmatrix}}{\begin{bmatrix}m_{0}\\m_{1}\\\vdots \\m_{k-1}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>G</mi> <mi>m</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>,</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd></mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd></mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(m)=Gm={\begin{bmatrix}1&0&0&\dots &0&g_{1,k+1}&\dots &g_{1,n}\\0&1&0&\dots &0&g_{2,k+1}&\dots &g_{2,n}\\0&0&1&\dots &0&g_{3,k+1}&\dots &g_{3,n}\\\vdots &\vdots &\vdots &&\vdots &\vdots &&\vdots \\0&\dots &0&\dots &1&g_{k,k+1}&\dots &g_{k,n}\end{bmatrix}}{\begin{bmatrix}m_{0}\\m_{1}\\\vdots \\m_{k-1}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa3eccc8781538855fd755ccc527960c162e89f9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.338ex; width:65.783ex; height:17.843ex;" alt="{\displaystyle C(m)=Gm={\begin{bmatrix}1&0&0&\dots &0&g_{1,k+1}&\dots &g_{1,n}\\0&1&0&\dots &0&g_{2,k+1}&\dots &g_{2,n}\\0&0&1&\dots &0&g_{3,k+1}&\dots &g_{3,n}\\\vdots &\vdots &\vdots &&\vdots &\vdots &&\vdots \\0&\dots &0&\dots &1&g_{k,k+1}&\dots &g_{k,n}\end{bmatrix}}{\begin{bmatrix}m_{0}\\m_{1}\\\vdots \\m_{k-1}\end{bmatrix}}}" /> <div class="mw-heading mw-heading4"><h4 id="Discrete_Fourier_transform_and_its_inverse">Discrete Fourier transform and its inverse</h4>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=11" title="Edit section: Discrete Fourier transform and its inverse">edit</a>]</div> A <a href="/wiki/Discrete_Fourier_transform_(general)" class="mw-redirect" title="Discrete Fourier transform (general)">discrete Fourier transform</a> is essentially the same as the encoding procedure; it uses the generator polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9fdc3d9c861e438bc344ab483c1e5a896fc002d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.934ex; height:2.009ex;" alt="{\displaystyle p_{m}}" /> to map a set of evaluation points into the message values as shown above: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(m)={\begin{bmatrix}p_{m}(a_{0})\\p_{m}(a_{1})\\\cdots \\p_{m}(a_{n-1})\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(m)={\begin{bmatrix}p_{m}(a_{0})\\p_{m}(a_{1})\\\cdots \\p_{m}(a_{n-1})\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afa4e3e90d19297cee876396a0015fcce94282f6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:21.769ex; height:12.843ex;" alt="{\displaystyle C(m)={\begin{bmatrix}p_{m}(a_{0})\\p_{m}(a_{1})\\\cdots \\p_{m}(a_{n-1})\end{bmatrix}}}" /> The inverse Fourier transform could be used to convert an error free set of n < q message values back into the encoding polynomial of k coefficients, with the constraint that in order for this to work, the set of evaluation points used to encode the message must be a set of increasing powers of α: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}=\alpha ^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}=\alpha ^{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/468962ee2e67a642a59cb13ec59b7bdeb8577aaf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.415ex; height:3.009ex;" alt="{\displaystyle a_{i}=\alpha ^{i}}" /> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0},\dots ,a_{n-1}=\{1,\alpha ,\alpha ^{2},\dots ,\alpha ^{n-1}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mi>α</mi> <mo>,</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0},\dots ,a_{n-1}=\{1,\alpha ,\alpha ^{2},\dots ,\alpha ^{n-1}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449d57e1ff3576a7392ca848ce4f2c6dc3b19f0b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.679ex; height:3.176ex;" alt="{\displaystyle a_{0},\dots ,a_{n-1}=\{1,\alpha ,\alpha ^{2},\dots ,\alpha ^{n-1}\}}" /> However, Lagrange interpolation performs the same conversion without the constraint on the set of evaluation points or the requirement of an error free set of message values and is used for systematic encoding, and in one of the steps of the <a href="#Gao_decoder">Gao decoder</a>. <div class="mw-heading mw-heading3"><h3 id="The_BCH_view:_The_codeword_as_a_sequence_of_coefficients">The BCH view: The codeword as a sequence of coefficients</h3>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=12" title="Edit section: The BCH view: The codeword as a sequence of coefficients">edit</a>]</div> In this view, the message is interpreted as the coefficients of a polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cb7afced134ef75572e5314a5d278c2d644f438" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.398ex; height:2.843ex;" alt="{\displaystyle p(x)}" />. The sender computes a related polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10f51eec88706f1a26ac7430dc71d92c15e71a6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.229ex; height:2.843ex;" alt="{\displaystyle s(x)}" /> of degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd0b0f32b28f51962943ee9ede4fb34198a2521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}" /> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\leq q-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≤</mo> <mi>q</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\leq q-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ac1dd7d24f9e6244c03477e6147963fcbaa3870" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.566ex; height:2.509ex;" alt="{\displaystyle n\leq q-1}" /> and sends the polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10f51eec88706f1a26ac7430dc71d92c15e71a6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.229ex; height:2.843ex;" alt="{\displaystyle s(x)}" />. The polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10f51eec88706f1a26ac7430dc71d92c15e71a6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.229ex; height:2.843ex;" alt="{\displaystyle s(x)}" /> is constructed by multiplying the message polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cb7afced134ef75572e5314a5d278c2d644f438" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.398ex; height:2.843ex;" alt="{\displaystyle p(x)}" />, which has degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21363ebd7038c93aae93127e7d910fc1b2e2c745" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.214ex; height:2.343ex;" alt="{\displaystyle k-1}" />, with a <a href="/wiki/Generator_polynomial" class="mw-redirect" title="Generator polynomial">generator polynomial</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}" /> of degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b98e1d6a69bccd09a4b9b69bdf03a08c1706c8c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.446ex; height:2.343ex;" alt="{\displaystyle n-k}" /> that is known to both the sender and the receiver. The generator polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}" /> is defined as the polynomial whose roots are sequential powers of the Galois field primitive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)=\left(x-\alpha ^{i}\right)\left(x-\alpha ^{i+1}\right)\cdots \left(x-\alpha ^{i+n-k-1}\right)=g_{0}+g_{1}x+\cdots +g_{n-k-1}x^{n-k-1}+x^{n-k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>⋯</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)=\left(x-\alpha ^{i}\right)\left(x-\alpha ^{i+1}\right)\cdots \left(x-\alpha ^{i+n-k-1}\right)=g_{0}+g_{1}x+\cdots +g_{n-k-1}x^{n-k-1}+x^{n-k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/266ba2246aba3426245ca8f4d37cc02674fdb737" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:86.47ex; height:3.343ex;" alt="{\displaystyle g(x)=\left(x-\alpha ^{i}\right)\left(x-\alpha ^{i+1}\right)\cdots \left(x-\alpha ^{i+n-k-1}\right)=g_{0}+g_{1}x+\cdots +g_{n-k-1}x^{n-k-1}+x^{n-k}}" /> For a "narrow sense code", <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fb857369fbc28dab4b5d76bb835e6f196b1d1cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.063ex; height:2.176ex;" alt="{\displaystyle i=1}" />. <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} =\left\{\left(s_{1},s_{2},\dots ,s_{n}\right)\;{\Big |}\;s(a)=\sum _{i=1}^{n}s_{i}a^{i}{\text{ is a polynomial that has at least the roots }}\alpha ^{1},\alpha ^{2},\dots ,\alpha ^{n-k}\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">|</mo> </mrow> </mrow> <mspace width="thickmathspace"></mspace> <mi>s</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mtext> is a polynomial that has at least the roots </mtext> </mrow> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msup> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {C} =\left\{\left(s_{1},s_{2},\dots ,s_{n}\right)\;{\Big |}\;s(a)=\sum _{i=1}^{n}s_{i}a^{i}{\text{ is a polynomial that has at least the roots }}\alpha ^{1},\alpha ^{2},\dots ,\alpha ^{n-k}\right\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/928e31a56ac14f355c33f34d783930bd035adb9f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:100.141ex; height:7.509ex;" alt="{\displaystyle \mathbf {C} =\left\{\left(s_{1},s_{2},\dots ,s_{n}\right)\;{\Big |}\;s(a)=\sum _{i=1}^{n}s_{i}a^{i}{\text{ is a polynomial that has at least the roots }}\alpha ^{1},\alpha ^{2},\dots ,\alpha ^{n-k}\right\}.}" /> <div class="mw-heading mw-heading4"><h4 id="Systematic_encoding_procedure">Systematic encoding procedure</h4>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=13" title="Edit section: Systematic encoding procedure">edit</a>]</div> The encoding procedure for the BCH view of Reed–Solomon codes can be modified to yield a <a href="/wiki/Systematic_code" title="Systematic code">systematic encoding procedure</a>, in which each codeword contains the message as a prefix, and simply appends error correcting symbols as a suffix. Here, instead of sending <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(x)=p(x)g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(x)=p(x)g(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3158d120df065bf2602fba044f9a93c886f6107c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.891ex; height:2.843ex;" alt="{\displaystyle s(x)=p(x)g(x)}" />, the encoder constructs the transmitted polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10f51eec88706f1a26ac7430dc71d92c15e71a6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.229ex; height:2.843ex;" alt="{\displaystyle s(x)}" /> such that the coefficients of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" /> largest monomials are equal to the corresponding coefficients of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cb7afced134ef75572e5314a5d278c2d644f438" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.398ex; height:2.843ex;" alt="{\displaystyle p(x)}" />, and the lower-order coefficients of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10f51eec88706f1a26ac7430dc71d92c15e71a6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.229ex; height:2.843ex;" alt="{\displaystyle s(x)}" /> are chosen exactly in such a way that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10f51eec88706f1a26ac7430dc71d92c15e71a6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.229ex; height:2.843ex;" alt="{\displaystyle s(x)}" /> becomes divisible by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}" />. Then the coefficients of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cb7afced134ef75572e5314a5d278c2d644f438" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.398ex; height:2.843ex;" alt="{\displaystyle p(x)}" /> are a subsequence of the coefficients of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10f51eec88706f1a26ac7430dc71d92c15e71a6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.229ex; height:2.843ex;" alt="{\displaystyle s(x)}" />. To get a code that is overall systematic, we construct the message polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cb7afced134ef75572e5314a5d278c2d644f438" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.398ex; height:2.843ex;" alt="{\displaystyle p(x)}" /> by interpreting the message as the sequence of its coefficients. Formally, the construction is done by multiplying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cb7afced134ef75572e5314a5d278c2d644f438" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.398ex; height:2.843ex;" alt="{\displaystyle p(x)}" /> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d2e395e2fd6847edf44e51eb4462b3d3047e4b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.156ex; height:2.509ex;" alt="{\displaystyle x^{t}}" /> to make room for the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=n-k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=n-k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20fd706000b382bebb1e22ada342754bce266745" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.384ex; height:2.343ex;" alt="{\displaystyle t=n-k}" /> check symbols, dividing that product by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}" /> to find the remainder, and then compensating for that remainder by subtracting it. The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" /> check symbols are created by computing the remainder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{r}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{r}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ad749cfd068bb4ac474dd6f7b3ddc581b408bee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.203ex; height:2.843ex;" alt="{\displaystyle s_{r}(x)}" />: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{r}(x)=p(x)\cdot x^{t}\ {\bmod {\ }}g(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> </mrow> </mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{r}(x)=p(x)\cdot x^{t}\ {\bmod {\ }}g(x).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f70be24efb641f4fef51128617c0af7b1c126cc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.189ex; height:3.009ex;" alt="{\displaystyle s_{r}(x)=p(x)\cdot x^{t}\ {\bmod {\ }}g(x).}" /> The remainder has degree at most <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a215d9553945bb84b3b5a79cc796fb7d6e0629f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.842ex; height:2.343ex;" alt="{\displaystyle t-1}" />, whereas the coefficients of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{t-1},x^{t-2},\dots ,x^{1},x^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{t-1},x^{t-2},\dots ,x^{1},x^{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5194c2d2b72bec9b6928529797f8cc2bd16f549" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.526ex; height:3.009ex;" alt="{\displaystyle x^{t-1},x^{t-2},\dots ,x^{1},x^{0}}" /> in the polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)\cdot x^{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(x)\cdot x^{t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e3a47cb5feaf5a7a7ff88c74fcb7b73add115ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.233ex; height:3.009ex;" alt="{\displaystyle p(x)\cdot x^{t}}" /> are zero. Therefore, the following definition of the codeword <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10f51eec88706f1a26ac7430dc71d92c15e71a6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.229ex; height:2.843ex;" alt="{\displaystyle s(x)}" /> has the property that the first <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" /> coefficients are identical to the coefficients of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cb7afced134ef75572e5314a5d278c2d644f438" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.398ex; height:2.843ex;" alt="{\displaystyle p(x)}" />: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(x)=p(x)\cdot x^{t}-s_{r}(x)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(x)=p(x)\cdot x^{t}-s_{r}(x)\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80b56eaeb0c6ba929f35c7b5a5faa50f684e1383" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.549ex; height:3.009ex;" alt="{\displaystyle s(x)=p(x)\cdot x^{t}-s_{r}(x)\,.}" /> As a result, the codewords <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10f51eec88706f1a26ac7430dc71d92c15e71a6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.229ex; height:2.843ex;" alt="{\displaystyle s(x)}" /> are indeed elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11de80478fce9090e43eed19100b37cc841661e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\displaystyle \mathbf {C} }" />, that is, they are divisible by the generator polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}" />:<a href="#cite_note-11">[10]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(x)\equiv p(x)\cdot x^{t}-s_{r}(x)\equiv s_{r}(x)-s_{r}(x)\equiv 0\mod g(x)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <mn>0</mn> <mspace width="1em"></mspace> <mi>mod</mi> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(x)\equiv p(x)\cdot x^{t}-s_{r}(x)\equiv s_{r}(x)-s_{r}(x)\equiv 0\mod g(x)\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27311fb25410046b604d40686d04de6da05304c5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.672ex; height:3.009ex;" alt="{\displaystyle s(x)\equiv p(x)\cdot x^{t}-s_{r}(x)\equiv s_{r}(x)-s_{r}(x)\equiv 0\mod g(x)\,.}" /> This function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" /> is a linear mapping. To generate the corresponding systematic encoding matrix G, set G's left square submatrix to the identity matrix and then encode each row: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G={\begin{bmatrix}1&0&0&\dots &0&g_{1,k+1}&\dots &g_{1,n}\\0&1&0&\dots &0&g_{2,k+1}&\dots &g_{2,n}\\0&0&1&\dots &0&g_{3,k+1}&\dots &g_{3,n}\\\vdots &\vdots &\vdots &&\vdots &\vdots &&\vdots \\0&\dots &0&\dots &1&g_{k,k+1}&\dots &g_{k,n}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>,</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd></mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd></mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G={\begin{bmatrix}1&0&0&\dots &0&g_{1,k+1}&\dots &g_{1,n}\\0&1&0&\dots &0&g_{2,k+1}&\dots &g_{2,n}\\0&0&1&\dots &0&g_{3,k+1}&\dots &g_{3,n}\\\vdots &\vdots &\vdots &&\vdots &\vdots &&\vdots \\0&\dots &0&\dots &1&g_{k,k+1}&\dots &g_{k,n}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30b871e5959a092f5e5921e0530bbd808abd466" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.338ex; width:45.946ex; height:17.843ex;" alt="{\displaystyle G={\begin{bmatrix}1&0&0&\dots &0&g_{1,k+1}&\dots &g_{1,n}\\0&1&0&\dots &0&g_{2,k+1}&\dots &g_{2,n}\\0&0&1&\dots &0&g_{3,k+1}&\dots &g_{3,n}\\\vdots &\vdots &\vdots &&\vdots &\vdots &&\vdots \\0&\dots &0&\dots &1&g_{k,k+1}&\dots &g_{k,n}\end{bmatrix}}}" /> Ignoring leading zeroes, the last row = <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}" />. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(m)=Gm}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>G</mi> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(m)=Gm}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff3a4737c628699d66b491532c2f8f6a370b5609" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.582ex; height:2.843ex;" alt="{\displaystyle C(m)=Gm}" /> for the following <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/861eac296135ef99f3b1b36f807616a9d69717f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.446ex; height:2.176ex;" alt="{\displaystyle n\times k}" />-matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" /> with elements from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" />: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(m)=Gm={\begin{bmatrix}1&0&0&\dots &0&g_{1,k+1}&\dots &g_{1,n}\\0&1&0&\dots &0&g_{2,k+1}&\dots &g_{2,n}\\0&0&1&\dots &0&g_{3,k+1}&\dots &g_{3,n}\\\vdots &\vdots &\vdots &&\vdots &\vdots &&\vdots \\0&\dots &0&\dots &1&g_{k,k+1}&\dots &g_{k,n}\end{bmatrix}}{\begin{bmatrix}m_{0}\\m_{1}\\\vdots \\m_{k-1}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>G</mi> <mi>m</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>,</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd></mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd></mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(m)=Gm={\begin{bmatrix}1&0&0&\dots &0&g_{1,k+1}&\dots &g_{1,n}\\0&1&0&\dots &0&g_{2,k+1}&\dots &g_{2,n}\\0&0&1&\dots &0&g_{3,k+1}&\dots &g_{3,n}\\\vdots &\vdots &\vdots &&\vdots &\vdots &&\vdots \\0&\dots &0&\dots &1&g_{k,k+1}&\dots &g_{k,n}\end{bmatrix}}{\begin{bmatrix}m_{0}\\m_{1}\\\vdots \\m_{k-1}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa3eccc8781538855fd755ccc527960c162e89f9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.338ex; width:65.783ex; height:17.843ex;" alt="{\displaystyle C(m)=Gm={\begin{bmatrix}1&0&0&\dots &0&g_{1,k+1}&\dots &g_{1,n}\\0&1&0&\dots &0&g_{2,k+1}&\dots &g_{2,n}\\0&0&1&\dots &0&g_{3,k+1}&\dots &g_{3,n}\\\vdots &\vdots &\vdots &&\vdots &\vdots &&\vdots \\0&\dots &0&\dots &1&g_{k,k+1}&\dots &g_{k,n}\end{bmatrix}}{\begin{bmatrix}m_{0}\\m_{1}\\\vdots \\m_{k-1}\end{bmatrix}}}" /> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=14" title="Edit section: Properties">edit</a>]</div> The Reed–Solomon code is a [n, k, n − k + 1] code; in other words, it is a <a href="/wiki/Linear_code" title="Linear code">linear block code</a> of length n (over F) with <a href="/wiki/Dimension_(vector_space)" title="Dimension (vector space)">dimension</a> k and minimum <a href="/wiki/Hamming_distance" title="Hamming distance">Hamming distance</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle d_{\min }=n-k+1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">min</mo> </mrow> </msub> <mo>=</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle d_{\min }=n-k+1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/979adaf88ea43ef25b999ba6ef10ca71ed1704d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.376ex; height:2.509ex;" alt="{\textstyle d_{\min }=n-k+1.}" /> The Reed–Solomon code is optimal in the sense that the minimum distance has the maximum value possible for a linear code of size (n, k); this is known as the <a href="/wiki/Singleton_bound" title="Singleton bound">Singleton bound</a>. Such a code is also called a <a href="/wiki/MDS_code" class="mw-redirect" title="MDS code">maximum distance separable (MDS) code</a>. The error-correcting ability of a Reed–Solomon code is determined by its minimum distance, or equivalently, by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b98e1d6a69bccd09a4b9b69bdf03a08c1706c8c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.446ex; height:2.343ex;" alt="{\displaystyle n-k}" />, the measure of redundancy in the block. If the locations of the error symbols are not known in advance, then a Reed–Solomon code can correct up to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n-k)/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n-k)/2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4e64a932c23c9c478ad0948a57c3496ade18057" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.581ex; height:2.843ex;" alt="{\displaystyle (n-k)/2}" /> erroneous symbols, i.e., it can correct half as many errors as there are redundant symbols added to the block. Sometimes error locations are known in advance (e.g., "side information" in <a href="/wiki/Demodulator" class="mw-redirect" title="Demodulator">demodulator</a> <a href="/wiki/Signal-to-noise_ratio" title="Signal-to-noise ratio">signal-to-noise ratios</a>)—these are called <a href="/wiki/Erasure_channel" class="mw-redirect" title="Erasure channel">erasures</a>. A Reed–Solomon code (like any <a href="/wiki/Maximum_distance_separable_code" class="mw-redirect" title="Maximum distance separable code">MDS code</a>) is able to correct twice as many erasures as errors, and any combination of errors and erasures can be corrected as long as the relation 2E + S ≤ n − k is satisfied, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" /> is the number of errors and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /> is the number of erasures in the block. <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:RS_BER.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/RS_BER.png/500px-RS_BER.png" decoding="async" width="400" height="299" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/6/6e/RS_BER.png 1.5x" data-file-width="561" data-file-height="420" /></a><figcaption> Theoretical BER performance of the Reed-Solomon code (N=255, K=233, QPSK, AWGN). Step-like characteristic.</figcaption></figure> The theoretical error bound can be described via the following formula for the <a href="/wiki/Additive_white_Gaussian_noise" title="Additive white Gaussian noise">AWGN</a> channel for <a href="/wiki/Frequency-shift_keying" title="Frequency-shift keying">FSK</a>:<a href="#cite_note-12">[11]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{b}\approx {\frac {2^{m-1}}{2^{m}-1}}{\frac {1}{n}}\sum _{\ell =t+1}^{n}\ell {n \choose \ell }P_{s}^{\ell }(1-P_{s})^{n-\ell }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> <mo>=</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>ℓ</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msubsup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>ℓ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{b}\approx {\frac {2^{m-1}}{2^{m}-1}}{\frac {1}{n}}\sum _{\ell =t+1}^{n}\ell {n \choose \ell }P_{s}^{\ell }(1-P_{s})^{n-\ell }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d138500c9968956444a24117bfa4f0ecb2f1ad1d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:40.884ex; height:7.009ex;" alt="{\displaystyle P_{b}\approx {\frac {2^{m-1}}{2^{m}-1}}{\frac {1}{n}}\sum _{\ell =t+1}^{n}\ell {n \choose \ell }P_{s}^{\ell }(1-P_{s})^{n-\ell }}" /> and for other modulation schemes: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{b}\approx {\frac {1}{m}}{\frac {1}{n}}\sum _{\ell =t+1}^{n}\ell {n \choose \ell }P_{s}^{\ell }(1-P_{s})^{n-\ell }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>m</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> <mo>=</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>ℓ</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msubsup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>ℓ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{b}\approx {\frac {1}{m}}{\frac {1}{n}}\sum _{\ell =t+1}^{n}\ell {n \choose \ell }P_{s}^{\ell }(1-P_{s})^{n-\ell }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67dc74149ab7efda5baeb04b3bbca256b2df9af7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:36.084ex; height:7.009ex;" alt="{\displaystyle P_{b}\approx {\frac {1}{m}}{\frac {1}{n}}\sum _{\ell =t+1}^{n}\ell {n \choose \ell }P_{s}^{\ell }(1-P_{s})^{n-\ell }}" /> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle t={\frac {1}{2}}(d_{\min }-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">min</mo> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle t={\frac {1}{2}}(d_{\min }-1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b707cd148c70fde8e2c23b39be4d1112958b83a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.59ex; height:3.509ex;" alt="{\textstyle t={\frac {1}{2}}(d_{\min }-1)}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{s}=1-(1-s)^{h}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>s</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{s}=1-(1-s)^{h}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea7a171226c887215c77db3da1337d66315a50a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.679ex; height:3.176ex;" alt="{\displaystyle P_{s}=1-(1-s)^{h}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h={\frac {m}{\log _{2}M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>M</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h={\frac {m}{\log _{2}M}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9f2a22c2938cc361136cfe8cea8764fda1e15c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.129ex; height:5.343ex;" alt="{\displaystyle h={\frac {m}{\log _{2}M}}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" /> is the symbol error rate in uncoded AWGN case and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" /> is the modulation order. For practical uses of Reed–Solomon codes, it is common to use a finite field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" /> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/667d0154f26e56e3f7979803f08afac16b4dcb16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.837ex; height:2.343ex;" alt="{\displaystyle 2^{m}}" /> elements. In this case, each symbol can be represented as an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" />-bit value. The sender sends the data points as encoded blocks, and the number of symbols in the encoded block is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=2^{m}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=2^{m}-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d67dc75750941f57f3231fdb1e7815a8b6976de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.334ex; height:2.509ex;" alt="{\displaystyle n=2^{m}-1}" />. Thus a Reed–Solomon code operating on 8-bit symbols has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=2^{8}-1=255}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>255</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=2^{8}-1=255}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0dd1e854fc9e10a4e1079fcaa5dd042ffb58d95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.299ex; height:2.843ex;" alt="{\displaystyle n=2^{8}-1=255}" /> symbols per block. (This is a very popular value because of the prevalence of <a href="/wiki/Byte-oriented" class="mw-redirect" title="Byte-oriented">byte-oriented</a> computer systems.) The number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" />, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k<n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo><</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k<n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48ab7000a6f47e3a09a79dcbe31b89272b0c1f1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.704ex; height:2.176ex;" alt="{\displaystyle k<n}" />, of data symbols in the block is a design parameter. A commonly used code encodes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=223}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>223</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=223}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9da7a76696da59aca7f65ee03fe3051c788bc5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.797ex; height:2.176ex;" alt="{\displaystyle k=223}" /> eight-bit data symbols plus 32 eight-bit parity symbols in an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=255}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>255</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=255}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bf778dbae8ef593cc3b7fc9918d600345f9761b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.981ex; height:2.176ex;" alt="{\displaystyle n=255}" />-symbol block; this is denoted as a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n,k)=(255,223)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>255</mn> <mo>,</mo> <mn>223</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n,k)=(255,223)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9082ff9055fc9735bf92b4a88b823c4ba9b10e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.366ex; height:2.843ex;" alt="{\displaystyle (n,k)=(255,223)}" /> code, and is capable of correcting up to 16 symbol errors per block. The Reed–Solomon code properties discussed above make them especially well-suited to applications where errors occur in <a href="/wiki/Burst_error" title="Burst error">bursts</a>. This is because it does not matter to the code how many bits in a symbol are in error — if multiple bits in a symbol are corrupted it only counts as a single error. Conversely, if a data stream is not characterized by error bursts or drop-outs but by random single bit errors, a Reed–Solomon code is usually a poor choice compared to a binary code. The Reed–Solomon code, like the <a href="/wiki/Convolutional_code" title="Convolutional code">convolutional code</a>, is a transparent code. This means that if the channel symbols have been <a href="/wiki/Bitwise_NOT" class="mw-redirect" title="Bitwise NOT">inverted</a> somewhere along the line, the decoders will still operate. The result will be the inversion of the original data. However, the Reed–Solomon code loses its transparency when the code is shortened (see 'Remarks' at the end of this section). The "missing" bits in a shortened code need to be filled by either zeros or ones, depending on whether the data is complemented or not. (To put it another way, if the symbols are inverted, then the zero-fill needs to be inverted to a one-fill.) For this reason it is mandatory that the sense of the data (i.e., true or complemented) be resolved before Reed–Solomon decoding. Whether the Reed–Solomon code is <a href="/wiki/Cyclic_code" title="Cyclic code">cyclic</a> or not depends on subtle details of the construction. In the original view of Reed and Solomon, where the codewords are the values of a polynomial, one can choose the sequence of evaluation points in such a way as to make the code cyclic. In particular, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /> is a <a href="/wiki/Primitive_root_modulo_n" title="Primitive root modulo n">primitive root</a> of the field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" />, then by definition all non-zero elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" /> take the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ^{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c114e2fb3c65eea83541fc81d81ab6df7fca4da2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.287ex; height:2.676ex;" alt="{\displaystyle \alpha ^{i}}" /> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in \{1,\dots ,q-1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>q</mi> <mo>−</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\in \{1,\dots ,q-1\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c3ac37485000c4839f665d057ac2f49eb9d4778" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.381ex; height:2.843ex;" alt="{\displaystyle i\in \{1,\dots ,q-1\}}" />, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=|F|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=|F|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed759e55114f652058cc4898f9f22171f13bc77d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.202ex; height:2.843ex;" alt="{\displaystyle q=|F|}" />. Each polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" /> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" /> gives rise to a codeword <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p(\alpha ^{1}),\dots ,p(\alpha ^{q-1}))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>p</mi> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p(\alpha ^{1}),\dots ,p(\alpha ^{q-1}))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51e92405b72f105b8406ba31290677ddcd45f0da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.063ex; height:3.176ex;" alt="{\displaystyle (p(\alpha ^{1}),\dots ,p(\alpha ^{q-1}))}" />. Since the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\mapsto p(\alpha a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">↦</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\mapsto p(\alpha a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acdc3864f3db55ce6427a6b03b3e368ff1031a42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.54ex; height:2.843ex;" alt="{\displaystyle a\mapsto p(\alpha a)}" /> is also a polynomial of the same degree, this function gives rise to a codeword <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p(\alpha ^{2}),\dots ,p(\alpha ^{q}))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>p</mi> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p(\alpha ^{2}),\dots ,p(\alpha ^{q}))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8a1192568a26a471b9beb731e9917109339bdf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.963ex; height:3.176ex;" alt="{\displaystyle (p(\alpha ^{2}),\dots ,p(\alpha ^{q}))}" />; since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{q}=\alpha ^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ^{q}=\alpha ^{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eabe1e3718440d8a01f878c6a9d10c5580c6efb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.116ex; height:2.676ex;" alt="{\displaystyle \alpha ^{q}=\alpha ^{1}}" /> holds, this codeword is the <a href="/wiki/Circular_shift" title="Circular shift">cyclic left-shift</a> of the original codeword derived from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" />. So choosing a sequence of primitive root powers as the evaluation points makes the original view Reed–Solomon code <a href="/wiki/Cyclic_code" title="Cyclic code">cyclic</a>. Reed–Solomon codes in the BCH view are always cyclic because <a href="/wiki/BCH_code#Properties" title="BCH code">BCH codes are cyclic</a>. <div class="mw-heading mw-heading3"><h3 id="Remarks">Remarks</h3>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=15" title="Edit section: Remarks">edit</a>]</div> Designers are not required to use the "natural" sizes of Reed–Solomon code blocks. A technique known as "shortening" can produce a smaller code of any desired size from a larger code. For example, the widely used (255,223) code can be converted to a (160,128) code by padding the unused portion of the source block with 95 binary zeroes and not transmitting them. At the decoder, the same portion of the block is loaded locally with binary zeroes. The QR code, Ver 3 (29×29) uses interleaved blocks. The message has 26 data bytes and is encoded using two Reed-Solomon code blocks. Each block is a (255,233) Reed Solomon code shortened to a (35,13) code. The Delsarte–Goethals–Seidel<a href="#cite_note-13">[12]</a> theorem illustrates an example of an application of shortened Reed–Solomon codes. In parallel to shortening, a technique known as <a href="/wiki/Punctured_code" title="Punctured code">puncturing</a> allows omitting some of the encoded parity symbols. <div class="mw-heading mw-heading2"><h2 id="BCH_view_decoders">BCH view decoders</h2>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=16" title="Edit section: BCH view decoders">edit</a>]</div> The decoders described in this section use the BCH view of a codeword as a sequence of coefficients. They use a fixed generator polynomial known to both encoder and decoder. <div class="mw-heading mw-heading3"><h3 id="Peterson–Gorenstein–Zierler_decoder">Peterson–Gorenstein–Zierler decoder</h3>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=17" title="Edit section: Peterson–Gorenstein–Zierler decoder">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/BCH_code#Peterson–Gorenstein–Zierler_algorithm" title="BCH code">Peterson–Gorenstein–Zierler algorithm</a></div> Daniel Gorenstein and Neal Zierler developed a decoder that was described in a MIT Lincoln Laboratory report by Zierler in January 1960 and later in a paper in June 1961.<a href="#cite_note-14">[13]</a><a href="#cite_note-15">[14]</a> The Gorenstein–Zierler decoder and the related work on BCH codes are described in the book Error Correcting Codes by <a href="/wiki/W._Wesley_Peterson" title="W. Wesley Peterson">W. Wesley Peterson</a> (1961).<a href="#cite_note-Peterson61-3">[3]</a>[<a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">page needed</a>] <div class="mw-heading mw-heading4"><h4 id="Formulation">Formulation</h4>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=18" title="Edit section: Formulation">edit</a>]</div> The transmitted message, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (c_{0},\ldots ,c_{i},\ldots ,c_{n-1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c_{0},\ldots ,c_{i},\ldots ,c_{n-1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52d0cf7091b1fc0ac3a6dad4068f6a0755eebfb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.359ex; height:2.843ex;" alt="{\displaystyle (c_{0},\ldots ,c_{i},\ldots ,c_{n-1})}" />, is viewed as the coefficients of a polynomial <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(x)=\sum _{i=0}^{n-1}c_{i}x^{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(x)=\sum _{i=0}^{n-1}c_{i}x^{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a3b03897ae82c02194a62a6c00e368ecea6bdb1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.653ex; height:7.343ex;" alt="{\displaystyle s(x)=\sum _{i=0}^{n-1}c_{i}x^{i}.}" /> As a result of the Reed–Solomon encoding procedure, s(x) is divisible by the generator polynomial <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)=\prod _{j=1}^{n-k}(x-\alpha ^{j}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)=\prod _{j=1}^{n-k}(x-\alpha ^{j}),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f53edd89685a7f34edd0881e2549f5e0b337e308" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:19.498ex; height:7.676ex;" alt="{\displaystyle g(x)=\prod _{j=1}^{n-k}(x-\alpha ^{j}),}" /> where α is a primitive element. Since s(x) is a multiple of the generator g(x), it follows that it "inherits" all its roots: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(x){\bmod {(}}x-\alpha ^{j})=g(x){\bmod {(}}x-\alpha ^{j})=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> </mrow> </mrow> <mi>x</mi> <mo>−</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> </mrow> </mrow> <mi>x</mi> <mo>−</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(x){\bmod {(}}x-\alpha ^{j})=g(x){\bmod {(}}x-\alpha ^{j})=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a8e945f31eb85d59f21d2ffe6b86e61a0a7a7f6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.606ex; height:3.176ex;" alt="{\displaystyle s(x){\bmod {(}}x-\alpha ^{j})=g(x){\bmod {(}}x-\alpha ^{j})=0.}" /> Therefore, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(\alpha ^{j})=0,\ j=1,2,\ldots ,n-k.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(\alpha ^{j})=0,\ j=1,2,\ldots ,n-k.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0db288d8ceb5602b9fb41d89f3191217a76c5e8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.859ex; height:3.176ex;" alt="{\displaystyle s(\alpha ^{j})=0,\ j=1,2,\ldots ,n-k.}" /> The transmitted polynomial is corrupted in transit by an error polynomial <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e(x)=\sum _{i=0}^{n-1}e_{i}x^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e(x)=\sum _{i=0}^{n-1}e_{i}x^{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8573e9892c332f594fa357c2c1775cf5697b2ff" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.076ex; height:7.343ex;" alt="{\displaystyle e(x)=\sum _{i=0}^{n-1}e_{i}x^{i}}" /> to produce the received polynomial <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(x)=s(x)+e(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>e</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r(x)=s(x)+e(x).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eac987597bc2bbf6acc21a72ffb4fd2768b4c1af" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.225ex; height:2.843ex;" alt="{\displaystyle r(x)=s(x)+e(x).}" /> Coefficient ei will be zero if there is no error at that power of x, and nonzero if there is an error. If there are ν errors at distinct powers ik of x, then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e(x)=\sum _{k=1}^{\nu }e_{i_{k}}x^{i_{k}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </munderover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e(x)=\sum _{k=1}^{\nu }e_{i_{k}}x^{i_{k}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebcda0ebd0b139ff4a100adca8e64822d3057049" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.442ex; height:6.843ex;" alt="{\displaystyle e(x)=\sum _{k=1}^{\nu }e_{i_{k}}x^{i_{k}}.}" /> The goal of the decoder is to find the number of errors (ν), the positions of the errors (ik), and the error values at those positions (eik). From those, e(x) can be calculated and subtracted from r(x) to get the originally sent message s(x). <div class="mw-heading mw-heading4"><h4 id="Syndrome_decoding">Syndrome decoding</h4>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=19" title="Edit section: Syndrome decoding">edit</a>]</div> The decoder starts by evaluating the polynomial as received at points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{1}\dots \alpha ^{n-k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>…</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ^{1}\dots \alpha ^{n-k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7884967ff259386ada548014852496717a4e5dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.88ex; height:2.676ex;" alt="{\displaystyle \alpha ^{1}\dots \alpha ^{n-k}}" />. We call the results of that evaluation the "syndromes" Sj. They are defined as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}S_{j}&=r(\alpha ^{j})=s(\alpha ^{j})+e(\alpha ^{j})=0+e(\alpha ^{j})\\&=e(\alpha ^{j})\\&=\sum _{k=1}^{\nu }e_{i_{k}}{(\alpha ^{j})}^{i_{k}},\quad j=1,2,\ldots ,n-k.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>s</mi> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <mi>e</mi> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>+</mo> <mi>e</mi> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mi>e</mi> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </munderover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msup> <mo>,</mo> <mspace width="1em"></mspace> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}S_{j}&=r(\alpha ^{j})=s(\alpha ^{j})+e(\alpha ^{j})=0+e(\alpha ^{j})\\&=e(\alpha ^{j})\\&=\sum _{k=1}^{\nu }e_{i_{k}}{(\alpha ^{j})}^{i_{k}},\quad j=1,2,\ldots ,n-k.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88184eeecbf1579025f134339c6271b827158ad2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:40.579ex; height:13.843ex;" alt="{\displaystyle {\begin{aligned}S_{j}&=r(\alpha ^{j})=s(\alpha ^{j})+e(\alpha ^{j})=0+e(\alpha ^{j})\\&=e(\alpha ^{j})\\&=\sum _{k=1}^{\nu }e_{i_{k}}{(\alpha ^{j})}^{i_{k}},\quad j=1,2,\ldots ,n-k.\end{aligned}}}" /> Note that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(\alpha ^{j})=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(\alpha ^{j})=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5747e401d1b507dbd824712b93bb6f8d720de55a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.558ex; height:3.176ex;" alt="{\displaystyle s(\alpha ^{j})=0}" /> because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10f51eec88706f1a26ac7430dc71d92c15e71a6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.229ex; height:2.843ex;" alt="{\displaystyle s(x)}" /> has roots at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ^{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27b185f9f9058e5b7f59374f9a0e2069f9d0c237" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.397ex; height:2.676ex;" alt="{\displaystyle \alpha ^{j}}" />, as shown in the previous section. The advantage of looking at the syndromes is that the message polynomial drops out. In other words, the syndromes only relate to the error and are unaffected by the actual contents of the message being transmitted. If the syndromes are all zero, the algorithm stops here and reports that the message was not corrupted in transit. <div class="mw-heading mw-heading4"><h4 id="Error_locators_and_error_values">Error locators and error values</h4>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=20" title="Edit section: Error locators and error values">edit</a>]</div> For convenience, define the error locators Xk and error values Yk as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{k}=\alpha ^{i_{k}},\quad Y_{k}=e_{i_{k}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msup> <mo>,</mo> <mspace width="1em"></mspace> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{k}=\alpha ^{i_{k}},\quad Y_{k}=e_{i_{k}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/230f97657fa0c9d19ba8c53d3ba51bd2b3d1f27e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.542ex; height:3.343ex;" alt="{\displaystyle X_{k}=\alpha ^{i_{k}},\quad Y_{k}=e_{i_{k}}.}" /> Then the syndromes can be written in terms of these error locators and error values as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{j}=\sum _{k=1}^{\nu }Y_{k}X_{k}^{j}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </munderover> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{j}=\sum _{k=1}^{\nu }Y_{k}X_{k}^{j}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f1b0d46a3a80a9f47d39ceb3cf0d17329a7eb0d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.274ex; height:6.843ex;" alt="{\displaystyle S_{j}=\sum _{k=1}^{\nu }Y_{k}X_{k}^{j}.}" /> This definition of the syndrome values is equivalent to the previous since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {(\alpha ^{j})}^{i_{k}}=\alpha ^{j\cdot i_{k}}={(\alpha ^{i_{k}})}^{j}=X_{k}^{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msup> <mo>=</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>⋅</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo>=</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {(\alpha ^{j})}^{i_{k}}=\alpha ^{j\cdot i_{k}}={(\alpha ^{i_{k}})}^{j}=X_{k}^{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07e0654db0632ff5ba8d86f4f796d3fbd1c22cf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.322ex; height:3.676ex;" alt="{\displaystyle {(\alpha ^{j})}^{i_{k}}=\alpha ^{j\cdot i_{k}}={(\alpha ^{i_{k}})}^{j}=X_{k}^{j}}" />. The syndromes give a system of n − k ≥ 2ν equations in 2ν unknowns, but that system of equations is nonlinear in the Xk and does not have an obvious solution. However, if the Xk were known (see below), then the syndrome equations provide a linear system of equations <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}X_{1}^{1}&X_{2}^{1}&\cdots &X_{\nu }^{1}\\X_{1}^{2}&X_{2}^{2}&\cdots &X_{\nu }^{2}\\\vdots &\vdots &\ddots &\vdots \\X_{1}^{n-k}&X_{2}^{n-k}&\cdots &X_{\nu }^{n-k}\\\end{bmatrix}}{\begin{bmatrix}Y_{1}\\Y_{2}\\\vdots \\Y_{\nu }\end{bmatrix}}={\begin{bmatrix}S_{1}\\S_{2}\\\vdots \\S_{n-k}\end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mtd> <mtd> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msubsup> </mtd> <mtd> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msubsup> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}X_{1}^{1}&X_{2}^{1}&\cdots &X_{\nu }^{1}\\X_{1}^{2}&X_{2}^{2}&\cdots &X_{\nu }^{2}\\\vdots &\vdots &\ddots &\vdots \\X_{1}^{n-k}&X_{2}^{n-k}&\cdots &X_{\nu }^{n-k}\\\end{bmatrix}}{\begin{bmatrix}Y_{1}\\Y_{2}\\\vdots \\Y_{\nu }\end{bmatrix}}={\begin{bmatrix}S_{1}\\S_{2}\\\vdots \\S_{n-k}\end{bmatrix}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6eb581cb88e71f26ef96c64ca0b75b420dc24574" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.005ex; width:48.532ex; height:15.176ex;" alt="{\displaystyle {\begin{bmatrix}X_{1}^{1}&X_{2}^{1}&\cdots &X_{\nu }^{1}\\X_{1}^{2}&X_{2}^{2}&\cdots &X_{\nu }^{2}\\\vdots &\vdots &\ddots &\vdots \\X_{1}^{n-k}&X_{2}^{n-k}&\cdots &X_{\nu }^{n-k}\\\end{bmatrix}}{\begin{bmatrix}Y_{1}\\Y_{2}\\\vdots \\Y_{\nu }\end{bmatrix}}={\begin{bmatrix}S_{1}\\S_{2}\\\vdots \\S_{n-k}\end{bmatrix}},}" /> which can easily be solved for the Yk error values. Consequently, the problem is finding the Xk, because then the leftmost matrix would be known, and both sides of the equation could be multiplied by its inverse, yielding Yk In the variant of this algorithm where the locations of the errors are already known (when it is being used as an <a href="/wiki/Erasure_code" title="Erasure code">erasure code</a>), this is the end. The error locations (Xk) are already known by some other method (for example, in an FM transmission, the sections where the bitstream was unclear or overcome with interference are probabilistically determinable from frequency analysis). In this scenario, up to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b98e1d6a69bccd09a4b9b69bdf03a08c1706c8c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.446ex; height:2.343ex;" alt="{\displaystyle n-k}" /> errors can be corrected. The rest of the algorithm serves to locate the errors and will require syndrome values up to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\nu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db2654d0632cb4b4cd30cfddb80a75ba7d743216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.395ex; height:2.176ex;" alt="{\displaystyle 2\nu }" />, instead of just the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" /> used thus far. This is why twice as many error-correcting symbols need to be added as can be corrected without knowing their locations. <div class="mw-heading mw-heading4"><h4 id="Error_locator_polynomial">Error locator polynomial</h4>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=21" title="Edit section: Error locator polynomial">edit</a>]</div> There is a linear recurrence relation that gives rise to a system of linear equations. Solving those equations identifies those error locations Xk. Define the error locator polynomial Λ(x) as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda (x)=\prod _{k=1}^{\nu }(1-xX_{k})=1+\Lambda _{1}x^{1}+\Lambda _{2}x^{2}+\cdots +\Lambda _{\nu }x^{\nu }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda (x)=\prod _{k=1}^{\nu }(1-xX_{k})=1+\Lambda _{1}x^{1}+\Lambda _{2}x^{2}+\cdots +\Lambda _{\nu }x^{\nu }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc71e84c5b681a0b93fda798dda52d6815f16179" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:55.219ex; height:6.843ex;" alt="{\displaystyle \Lambda (x)=\prod _{k=1}^{\nu }(1-xX_{k})=1+\Lambda _{1}x^{1}+\Lambda _{2}x^{2}+\cdots +\Lambda _{\nu }x^{\nu }.}" /> The zeros of Λ(x) are the reciprocals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{k}^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{k}^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d660f988c86d5e177544cf19b5a1d46bb526202" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.33ex; height:3.343ex;" alt="{\displaystyle X_{k}^{-1}}" />. This follows from the above product notation construction, since if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=X_{k}^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=X_{k}^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12ff650063a65b2edc9a81c3c4ce5710446c3ee8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.758ex; height:3.343ex;" alt="{\displaystyle x=X_{k}^{-1}}" />, then one of the multiplied terms will be zero, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1-X_{k}^{-1}\cdot X_{k})=1-1=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>⋅</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1-X_{k}^{-1}\cdot X_{k})=1-1=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e88006aa912ddfb3855c42cc6c1bd8a5cf8dd974" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.359ex; height:3.343ex;" alt="{\displaystyle (1-X_{k}^{-1}\cdot X_{k})=1-1=0}" />, making the whole polynomial evaluate to zero: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda (X_{k}^{-1})=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda (X_{k}^{-1})=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57d5a1e8df5a2ee5051b52ebec33a7916cdb2862" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.66ex; height:3.343ex;" alt="{\displaystyle \Lambda (X_{k}^{-1})=0.}" /> Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" /> be any integer such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq j\leq \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>j</mi> <mo>≤</mo> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq j\leq \nu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fd9b9bdc56251f66bea3a5914cb6867abbd458a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.55ex; height:2.509ex;" alt="{\displaystyle 1\leq j\leq \nu }" />. Multiply both sides by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{k}X_{k}^{j+\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mi>ν</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{k}X_{k}^{j+\nu }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ec8abe7c475b5a73359c3993aae262022f5e70b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.496ex; height:3.509ex;" alt="{\displaystyle Y_{k}X_{k}^{j+\nu }}" />, and it will still be zero: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&Y_{k}X_{k}^{j+\nu }\Lambda (X_{k}^{-1})=0,\\&Y_{k}X_{k}^{j+\nu }(1+\Lambda _{1}X_{k}^{-1}+\Lambda _{2}X_{k}^{-2}+\cdots +\Lambda _{\nu }X_{k}^{-\nu })=0,\\&Y_{k}X_{k}^{j+\nu }+\Lambda _{1}Y_{k}X_{k}^{j+\nu }X_{k}^{-1}+\Lambda _{2}Y_{k}X_{k}^{j+\nu }X_{k}^{-2}+\cdots +\Lambda _{\nu }Y_{k}X_{k}^{j+\nu }X_{k}^{-\nu }=0,\\&Y_{k}X_{k}^{j+\nu }+\Lambda _{1}Y_{k}X_{k}^{j+\nu -1}+\Lambda _{2}Y_{k}X_{k}^{j+\nu -2}+\cdots +\Lambda _{\nu }Y_{k}X_{k}^{j}=0.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd></mtd> <mtd> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mi>ν</mi> </mrow> </msubsup> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mi>ν</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>ν</mi> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mi>ν</mi> </mrow> </msubsup> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mi>ν</mi> </mrow> </msubsup> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mi>ν</mi> </mrow> </msubsup> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mi>ν</mi> </mrow> </msubsup> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>ν</mi> </mrow> </msubsup> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mi>ν</mi> </mrow> </msubsup> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mi>ν</mi> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mi>ν</mi> <mo>−</mo> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msubsup> <mo>=</mo> <mn>0.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&Y_{k}X_{k}^{j+\nu }\Lambda (X_{k}^{-1})=0,\\&Y_{k}X_{k}^{j+\nu }(1+\Lambda _{1}X_{k}^{-1}+\Lambda _{2}X_{k}^{-2}+\cdots +\Lambda _{\nu }X_{k}^{-\nu })=0,\\&Y_{k}X_{k}^{j+\nu }+\Lambda _{1}Y_{k}X_{k}^{j+\nu }X_{k}^{-1}+\Lambda _{2}Y_{k}X_{k}^{j+\nu }X_{k}^{-2}+\cdots +\Lambda _{\nu }Y_{k}X_{k}^{j+\nu }X_{k}^{-\nu }=0,\\&Y_{k}X_{k}^{j+\nu }+\Lambda _{1}Y_{k}X_{k}^{j+\nu -1}+\Lambda _{2}Y_{k}X_{k}^{j+\nu -2}+\cdots +\Lambda _{\nu }Y_{k}X_{k}^{j}=0.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb0c988668bad414b19c38b117012d629303ef50" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.475ex; margin-bottom: -0.197ex; width:70.816ex; height:14.509ex;" alt="{\displaystyle {\begin{aligned}&Y_{k}X_{k}^{j+\nu }\Lambda (X_{k}^{-1})=0,\\&Y_{k}X_{k}^{j+\nu }(1+\Lambda _{1}X_{k}^{-1}+\Lambda _{2}X_{k}^{-2}+\cdots +\Lambda _{\nu }X_{k}^{-\nu })=0,\\&Y_{k}X_{k}^{j+\nu }+\Lambda _{1}Y_{k}X_{k}^{j+\nu }X_{k}^{-1}+\Lambda _{2}Y_{k}X_{k}^{j+\nu }X_{k}^{-2}+\cdots +\Lambda _{\nu }Y_{k}X_{k}^{j+\nu }X_{k}^{-\nu }=0,\\&Y_{k}X_{k}^{j+\nu }+\Lambda _{1}Y_{k}X_{k}^{j+\nu -1}+\Lambda _{2}Y_{k}X_{k}^{j+\nu -2}+\cdots +\Lambda _{\nu }Y_{k}X_{k}^{j}=0.\end{aligned}}}" /> Sum for k = 1 to ν, and it will still be zero: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=1}^{\nu }(Y_{k}X_{k}^{j+\nu }+\Lambda _{1}Y_{k}X_{k}^{j+\nu -1}+\Lambda _{2}Y_{k}X_{k}^{j+\nu -2}+\cdots +\Lambda _{\nu }Y_{k}X_{k}^{j})=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mi>ν</mi> </mrow> </msubsup> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mi>ν</mi> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mi>ν</mi> <mo>−</mo> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{\nu }(Y_{k}X_{k}^{j+\nu }+\Lambda _{1}Y_{k}X_{k}^{j+\nu -1}+\Lambda _{2}Y_{k}X_{k}^{j+\nu -2}+\cdots +\Lambda _{\nu }Y_{k}X_{k}^{j})=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2b47e3add4cdc8e2cdaaf76db3efebf0f5ccf59" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:64.348ex; height:6.843ex;" alt="{\displaystyle \sum _{k=1}^{\nu }(Y_{k}X_{k}^{j+\nu }+\Lambda _{1}Y_{k}X_{k}^{j+\nu -1}+\Lambda _{2}Y_{k}X_{k}^{j+\nu -2}+\cdots +\Lambda _{\nu }Y_{k}X_{k}^{j})=0.}" /> Collect each term into its own sum: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\sum _{k=1}^{\nu }Y_{k}X_{k}^{j+\nu }\right)+\left(\sum _{k=1}^{\nu }\Lambda _{1}Y_{k}X_{k}^{j+\nu -1}\right)+\left(\sum _{k=1}^{\nu }\Lambda _{2}Y_{k}X_{k}^{j+\nu -2}\right)+\cdots +\left(\sum _{k=1}^{\nu }\Lambda _{\nu }Y_{k}X_{k}^{j}\right)=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </munderover> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mi>ν</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </munderover> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mi>ν</mi> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </munderover> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mi>ν</mi> <mo>−</mo> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </munderover> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\sum _{k=1}^{\nu }Y_{k}X_{k}^{j+\nu }\right)+\left(\sum _{k=1}^{\nu }\Lambda _{1}Y_{k}X_{k}^{j+\nu -1}\right)+\left(\sum _{k=1}^{\nu }\Lambda _{2}Y_{k}X_{k}^{j+\nu -2}\right)+\cdots +\left(\sum _{k=1}^{\nu }\Lambda _{\nu }Y_{k}X_{k}^{j}\right)=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58c5bcd1abf51b3916c7fe0a8439db419a284df0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:88.878ex; height:7.509ex;" alt="{\displaystyle \left(\sum _{k=1}^{\nu }Y_{k}X_{k}^{j+\nu }\right)+\left(\sum _{k=1}^{\nu }\Lambda _{1}Y_{k}X_{k}^{j+\nu -1}\right)+\left(\sum _{k=1}^{\nu }\Lambda _{2}Y_{k}X_{k}^{j+\nu -2}\right)+\cdots +\left(\sum _{k=1}^{\nu }\Lambda _{\nu }Y_{k}X_{k}^{j}\right)=0.}" /> Extract the constant values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac0a4a98a414e3480335f9ba652d12571ec6733" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }" /> that are unaffected by the summation: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\sum _{k=1}^{\nu }Y_{k}X_{k}^{j+\nu }\right)+\Lambda _{1}\left(\sum _{k=1}^{\nu }Y_{k}X_{k}^{j+\nu -1}\right)+\Lambda _{2}\left(\sum _{k=1}^{\nu }Y_{k}X_{k}^{j+\nu -2}\right)+\cdots +\Lambda _{\nu }\left(\sum _{k=1}^{\nu }Y_{k}X_{k}^{j}\right)=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </munderover> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mi>ν</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </munderover> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mi>ν</mi> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </munderover> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mi>ν</mi> <mo>−</mo> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </munderover> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\sum _{k=1}^{\nu }Y_{k}X_{k}^{j+\nu }\right)+\Lambda _{1}\left(\sum _{k=1}^{\nu }Y_{k}X_{k}^{j+\nu -1}\right)+\Lambda _{2}\left(\sum _{k=1}^{\nu }Y_{k}X_{k}^{j+\nu -2}\right)+\cdots +\Lambda _{\nu }\left(\sum _{k=1}^{\nu }Y_{k}X_{k}^{j}\right)=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6946837d89e53f1a059c998262f7e68feefea948" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:90.039ex; height:7.509ex;" alt="{\displaystyle \left(\sum _{k=1}^{\nu }Y_{k}X_{k}^{j+\nu }\right)+\Lambda _{1}\left(\sum _{k=1}^{\nu }Y_{k}X_{k}^{j+\nu -1}\right)+\Lambda _{2}\left(\sum _{k=1}^{\nu }Y_{k}X_{k}^{j+\nu -2}\right)+\cdots +\Lambda _{\nu }\left(\sum _{k=1}^{\nu }Y_{k}X_{k}^{j}\right)=0.}" /> These summations are now equivalent to the syndrome values, which we know and can substitute in. This therefore reduces to <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{j+\nu }+\Lambda _{1}S_{j+\nu -1}+\cdots +\Lambda _{\nu -1}S_{j+1}+\Lambda _{\nu }S_{j}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mi>ν</mi> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mi>ν</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{j+\nu }+\Lambda _{1}S_{j+\nu -1}+\cdots +\Lambda _{\nu -1}S_{j+1}+\Lambda _{\nu }S_{j}=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b47e0311e202381d7255be57a2e79802a3fbaddc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:47.033ex; height:2.843ex;" alt="{\displaystyle S_{j+\nu }+\Lambda _{1}S_{j+\nu -1}+\cdots +\Lambda _{\nu -1}S_{j+1}+\Lambda _{\nu }S_{j}=0.}" /> Subtracting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{j+\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{j+\nu }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/659264950ad95b0cbeba252514ccc41acdc47bc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.484ex; height:2.843ex;" alt="{\displaystyle S_{j+\nu }}" /> from both sides yields <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{j}\Lambda _{\nu }+S_{j+1}\Lambda _{\nu -1}+\cdots +S_{j+\nu -1}\Lambda _{1}=-S_{j+\nu }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mi>ν</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mi>ν</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{j}\Lambda _{\nu }+S_{j+1}\Lambda _{\nu -1}+\cdots +S_{j+\nu -1}\Lambda _{1}=-S_{j+\nu }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9f847ca88038b86ea92885098d3e34edefc5054" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:44.838ex; height:2.843ex;" alt="{\displaystyle S_{j}\Lambda _{\nu }+S_{j+1}\Lambda _{\nu -1}+\cdots +S_{j+\nu -1}\Lambda _{1}=-S_{j+\nu }.}" /> Recall that j was chosen to be any integer between 1 and v inclusive, and this equivalence is true for all such values. Therefore, we have v linear equations, not just one. This system of linear equations can therefore be solved for the coefficients Λi of the error-location polynomial: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}S_{1}&S_{2}&\cdots &S_{\nu }\\S_{2}&S_{3}&\cdots &S_{\nu +1}\\\vdots &\vdots &\ddots &\vdots \\S_{\nu }&S_{\nu +1}&\cdots &S_{2\nu -1}\end{bmatrix}}{\begin{bmatrix}\Lambda _{\nu }\\\Lambda _{\nu -1}\\\vdots \\\Lambda _{1}\end{bmatrix}}={\begin{bmatrix}-S_{\nu +1}\\-S_{\nu +2}\\\vdots \\-S_{\nu +\nu }\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>ν</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> <mo>+</mo> <mi>ν</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}S_{1}&S_{2}&\cdots &S_{\nu }\\S_{2}&S_{3}&\cdots &S_{\nu +1}\\\vdots &\vdots &\ddots &\vdots \\S_{\nu }&S_{\nu +1}&\cdots &S_{2\nu -1}\end{bmatrix}}{\begin{bmatrix}\Lambda _{\nu }\\\Lambda _{\nu -1}\\\vdots \\\Lambda _{1}\end{bmatrix}}={\begin{bmatrix}-S_{\nu +1}\\-S_{\nu +2}\\\vdots \\-S_{\nu +\nu }\end{bmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2653662fab6b37b318d021d7d56bb4ee067d7b3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:49.16ex; height:14.176ex;" alt="{\displaystyle {\begin{bmatrix}S_{1}&S_{2}&\cdots &S_{\nu }\\S_{2}&S_{3}&\cdots &S_{\nu +1}\\\vdots &\vdots &\ddots &\vdots \\S_{\nu }&S_{\nu +1}&\cdots &S_{2\nu -1}\end{bmatrix}}{\begin{bmatrix}\Lambda _{\nu }\\\Lambda _{\nu -1}\\\vdots \\\Lambda _{1}\end{bmatrix}}={\begin{bmatrix}-S_{\nu +1}\\-S_{\nu +2}\\\vdots \\-S_{\nu +\nu }\end{bmatrix}}.}" /> The above assumes that the decoder knows the number of errors ν, but that number has not been determined yet. The PGZ decoder does not determine ν directly but rather searches for it by trying successive values. The decoder first assumes the largest value for a trial ν and sets up the linear system for that value. If the equations can be solved (i.e., the matrix determinant is nonzero), then that trial value is the number of errors. If the linear system cannot be solved, then the trial ν is reduced by one and the next smaller system is examined.<a href="#cite_note-16">[15]</a> <div class="mw-heading mw-heading4"><h4 id="Find_the_roots_of_the_error_locator_polynomial">Find the roots of the error locator polynomial</h4>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=22" title="Edit section: Find the roots of the error locator polynomial">edit</a>]</div> Use the coefficients Λi found in the last step to build the error location polynomial. The roots of the error location polynomial can be found by exhaustive search. The error locators Xk are the reciprocals of those roots. The order of coefficients of the error location polynomial can be reversed, in which case the roots of that reversed polynomial are the error locators <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33c25229c6989c235f9cbb7908331f6d01d0abfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.013ex; height:2.509ex;" alt="{\displaystyle X_{k}}" /> (not their reciprocals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{k}^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{k}^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d660f988c86d5e177544cf19b5a1d46bb526202" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.33ex; height:3.343ex;" alt="{\displaystyle X_{k}^{-1}}" />). <a href="/wiki/Chien_search" title="Chien search">Chien search</a> is an efficient implementation of this step. <div class="mw-heading mw-heading4"><h4 id="Calculate_the_error_values">Calculate the error values</h4>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=23" title="Edit section: Calculate the error values">edit</a>]</div> Once the error locators Xk are known, the error values can be determined. This can be done by direct solution for Yk in the <a href="#Error_locators_and_error_values">error equations</a> matrix given above, or using the <a href="/wiki/Forney_algorithm" title="Forney algorithm">Forney algorithm</a>. <div class="mw-heading mw-heading4"><h4 id="Calculate_the_error_locations">Calculate the error locations</h4>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=24" title="Edit section: Calculate the error locations">edit</a>]</div> Calculate ik by taking the log base <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /> of Xk. This is generally done using a precomputed lookup table. <div class="mw-heading mw-heading4"><h4 id="Fix_the_errors">Fix the errors</h4>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=25" title="Edit section: Fix the errors">edit</a>]</div> Finally, e(x) is generated from ik and eik and then is subtracted from r(x) to get the originally sent message s(x), with errors corrected. <div class="mw-heading mw-heading4"><h4 id="Example">Example</h4>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=26" title="Edit section: Example">edit</a>]</div> Consider the Reed–Solomon code defined in GF(929) with α = 3 and t = 4 (this is used in <a href="/wiki/PDF417" title="PDF417">PDF417</a> barcodes) for a RS(7,3) code. The generator polynomial is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)=(x-3)(x-3^{2})(x-3^{3})(x-3^{4})=x^{4}+809x^{3}+723x^{2}+568x+522.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>809</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>723</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>568</mn> <mi>x</mi> <mo>+</mo> <mn>522.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)=(x-3)(x-3^{2})(x-3^{3})(x-3^{4})=x^{4}+809x^{3}+723x^{2}+568x+522.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f2514b3b52bd5b92bf59068fb4c80689ba3bdac" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:76.621ex; height:3.176ex;" alt="{\displaystyle g(x)=(x-3)(x-3^{2})(x-3^{3})(x-3^{4})=x^{4}+809x^{3}+723x^{2}+568x+522.}" /> If the message polynomial is p(x) = 3 x2 + 2 x + 1, then a systematic codeword is encoded as follows: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{r}(x)=p(x)\,x^{t}{\bmod {g}}(x)=547x^{3}+738x^{2}+442x+455,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>547</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>738</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>442</mn> <mi>x</mi> <mo>+</mo> <mn>455</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{r}(x)=p(x)\,x^{t}{\bmod {g}}(x)=547x^{3}+738x^{2}+442x+455,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8313940080cc75627d52234757db415e1b5864a6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.402ex; height:3.176ex;" alt="{\displaystyle s_{r}(x)=p(x)\,x^{t}{\bmod {g}}(x)=547x^{3}+738x^{2}+442x+455,}" /> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(x)=p(x)\,x^{t}-s_{r}(x)=3x^{6}+2x^{5}+1x^{4}+382x^{3}+191x^{2}+487x+474.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>382</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>191</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>487</mn> <mi>x</mi> <mo>+</mo> <mn>474.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(x)=p(x)\,x^{t}-s_{r}(x)=3x^{6}+2x^{5}+1x^{4}+382x^{3}+191x^{2}+487x+474.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09a1150b2b337240627d1d9231371e28566b1dc9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:73.696ex; height:3.176ex;" alt="{\displaystyle s(x)=p(x)\,x^{t}-s_{r}(x)=3x^{6}+2x^{5}+1x^{4}+382x^{3}+191x^{2}+487x+474.}" /> Errors in transmission might cause this to be received instead: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(x)=s(x)+e(x)=3x^{6}+2x^{5}+123x^{4}+456x^{3}+191x^{2}+487x+474.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>e</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mn>123</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>456</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>191</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>487</mn> <mi>x</mi> <mo>+</mo> <mn>474.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r(x)=s(x)+e(x)=3x^{6}+2x^{5}+123x^{4}+456x^{3}+191x^{2}+487x+474.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e6c4b82e155f45b2ef501e6190199def0aaa96d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:72.377ex; height:3.176ex;" alt="{\displaystyle r(x)=s(x)+e(x)=3x^{6}+2x^{5}+123x^{4}+456x^{3}+191x^{2}+487x+474.}" /> The syndromes are calculated by evaluating r at powers of α: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1}=r(3^{1})=3\cdot 3^{6}+2\cdot 3^{5}+123\cdot 3^{4}+456\cdot 3^{3}+191\cdot 3^{2}+487\cdot 3+474=732,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mn>3</mn> <mo>⋅</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mo>⋅</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mn>123</mn> <mo>⋅</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>456</mn> <mo>⋅</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>191</mn> <mo>⋅</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>487</mn> <mo>⋅</mo> <mn>3</mn> <mo>+</mo> <mn>474</mn> <mo>=</mo> <mn>732</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{1}=r(3^{1})=3\cdot 3^{6}+2\cdot 3^{5}+123\cdot 3^{4}+456\cdot 3^{3}+191\cdot 3^{2}+487\cdot 3+474=732,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b8d0f1a3c7dfa168b590442f2a5e151c313c0dd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:80.108ex; height:3.176ex;" alt="{\displaystyle S_{1}=r(3^{1})=3\cdot 3^{6}+2\cdot 3^{5}+123\cdot 3^{4}+456\cdot 3^{3}+191\cdot 3^{2}+487\cdot 3+474=732,}" /> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{2}=r(3^{2})=637,\quad S_{3}=r(3^{3})=762,\quad S_{4}=r(3^{4})=925,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mn>637</mn> <mo>,</mo> <mspace width="1em"></mspace> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mn>762</mn> <mo>,</mo> <mspace width="1em"></mspace> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mn>925</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{2}=r(3^{2})=637,\quad S_{3}=r(3^{3})=762,\quad S_{4}=r(3^{4})=925,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54861f36824a7bbb4bc62bc5251de0400de9993d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:59.074ex; height:3.176ex;" alt="{\displaystyle S_{2}=r(3^{2})=637,\quad S_{3}=r(3^{3})=762,\quad S_{4}=r(3^{4})=925,}" /> yielding the system <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}732&637\\637&762\end{bmatrix}}{\begin{bmatrix}\Lambda _{2}\\\Lambda _{1}\end{bmatrix}}={\begin{bmatrix}-762\\-925\end{bmatrix}}={\begin{bmatrix}167\\004\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>732</mn> </mtd> <mtd> <mn>637</mn> </mtd> </mtr> <mtr> <mtd> <mn>637</mn> </mtd> <mtd> <mn>762</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>762</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>925</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>167</mn> </mtd> </mtr> <mtr> <mtd> <mn>004</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}732&637\\637&762\end{bmatrix}}{\begin{bmatrix}\Lambda _{2}\\\Lambda _{1}\end{bmatrix}}={\begin{bmatrix}-762\\-925\end{bmatrix}}={\begin{bmatrix}167\\004\end{bmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33d57ed506100e179e833891349e5be1d8c80e99" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.417ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}732&637\\637&762\end{bmatrix}}{\begin{bmatrix}\Lambda _{2}\\\Lambda _{1}\end{bmatrix}}={\begin{bmatrix}-762\\-925\end{bmatrix}}={\begin{bmatrix}167\\004\end{bmatrix}}.}" /> Using <a href="/wiki/Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a>, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}001&000\\000&001\end{bmatrix}}{\begin{bmatrix}\Lambda _{2}\\\Lambda _{1}\end{bmatrix}}={\begin{bmatrix}329\\821\end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>001</mn> </mtd> <mtd> <mn>000</mn> </mtd> </mtr> <mtr> <mtd> <mn>000</mn> </mtd> <mtd> <mn>001</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>329</mn> </mtd> </mtr> <mtr> <mtd> <mn>821</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}001&000\\000&001\end{bmatrix}}{\begin{bmatrix}\Lambda _{2}\\\Lambda _{1}\end{bmatrix}}={\begin{bmatrix}329\\821\end{bmatrix}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f28bf8c1ba3648fd8a9a5587c2776244b8b6a418" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.817ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}001&000\\000&001\end{bmatrix}}{\begin{bmatrix}\Lambda _{2}\\\Lambda _{1}\end{bmatrix}}={\begin{bmatrix}329\\821\end{bmatrix}},}" /> so <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda (x)=329x^{2}+821x+001,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>329</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>821</mn> <mi>x</mi> <mo>+</mo> <mn>001</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda (x)=329x^{2}+821x+001,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c66bd001c5521e6bbba9412486ee1f2ae585c7a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.354ex; height:3.176ex;" alt="{\displaystyle \Lambda (x)=329x^{2}+821x+001,}" /> with roots x1 = 757 = 3−3 and x2 = 562 = 3−4. The coefficients can be reversed: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(x)=001x^{2}+821x+329,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>001</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>821</mn> <mi>x</mi> <mo>+</mo> <mn>329</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(x)=001x^{2}+821x+329,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e8001cebb693712e871fec865e83a295f0d1125" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.505ex; height:3.176ex;" alt="{\displaystyle R(x)=001x^{2}+821x+329,}" /> to produce roots 27 = 33 and 81 = 34 with positive exponents, but typically this isn't used. The logarithm of the inverted roots corresponds to the error locations (right to left, location 0 is the last term in the codeword). To calculate the error values, apply the <a href="/wiki/Forney_algorithm" title="Forney algorithm">Forney algorithm</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega (x)=S(x)\Lambda (x){\bmod {x}}^{4}=546x+732,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <mn>546</mn> <mi>x</mi> <mo>+</mo> <mn>732</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega (x)=S(x)\Lambda (x){\bmod {x}}^{4}=546x+732,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb1a9ce5f00064032eb94974d40a0ebe4d881271" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.261ex; height:3.176ex;" alt="{\displaystyle \Omega (x)=S(x)\Lambda (x){\bmod {x}}^{4}=546x+732,}" /> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda '(x)=658x+821,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Λ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>658</mn> <mi>x</mi> <mo>+</mo> <mn>821</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda '(x)=658x+821,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e69ccd74df3eac52b38caab416dd9c44931ae68c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.327ex; height:3.009ex;" alt="{\displaystyle \Lambda '(x)=658x+821,}" /> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{1}=-\Omega (x_{1})/\Lambda '(x_{1})=074,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi mathvariant="normal">Λ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>074</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{1}=-\Omega (x_{1})/\Lambda '(x_{1})=074,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/183a7cf7994678bc5a5c7cce36a4108f995a4f75" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.802ex; height:3.009ex;" alt="{\displaystyle e_{1}=-\Omega (x_{1})/\Lambda '(x_{1})=074,}" /> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{2}=-\Omega (x_{2})/\Lambda '(x_{2})=122.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi mathvariant="normal">Λ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>122.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{2}=-\Omega (x_{2})/\Lambda '(x_{2})=122.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a3a86f7e53941c95f3c08fa5f77dd2ee23a9b1d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.802ex; height:3.009ex;" alt="{\displaystyle e_{2}=-\Omega (x_{2})/\Lambda '(x_{2})=122.}" /> Subtracting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{1}x^{3}+e_{2}x^{4}=74x^{3}+122x^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <mn>74</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>122</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{1}x^{3}+e_{2}x^{4}=74x^{3}+122x^{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21d55fb8e447b2ab77be924dfe23473294aa5638" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.403ex; height:3.009ex;" alt="{\displaystyle e_{1}x^{3}+e_{2}x^{4}=74x^{3}+122x^{4}}" /> from the received polynomial r(x) reproduces the original codeword s. <div class="mw-heading mw-heading3"><h3 id="Berlekamp–Massey_decoder">Berlekamp–Massey decoder</h3>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=27" title="Edit section: Berlekamp–Massey decoder">edit</a>]</div> The <a href="/wiki/Berlekamp%E2%80%93Massey_algorithm" title="Berlekamp–Massey algorithm">Berlekamp–Massey algorithm</a> is an alternate iterative procedure for finding the error locator polynomial. During each iteration, it calculates a discrepancy based on a current instance of Λ(x) with an assumed number of errors e: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta =S_{i}+\Lambda _{1}\ S_{i-1}+\cdots +\Lambda _{e}\ S_{i-e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mtext> </mtext> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mtext> </mtext> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mi>e</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta =S_{i}+\Lambda _{1}\ S_{i-1}+\cdots +\Lambda _{e}\ S_{i-e}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5311791419d9158c0c9ccd62b879722d94f2d5b3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.538ex; height:2.509ex;" alt="{\displaystyle \Delta =S_{i}+\Lambda _{1}\ S_{i-1}+\cdots +\Lambda _{e}\ S_{i-e}}" /> and then adjusts Λ(x) and e so that a recalculated Δ would be zero. The article <a href="/wiki/Berlekamp%E2%80%93Massey_algorithm" title="Berlekamp–Massey algorithm">Berlekamp–Massey algorithm</a> has a detailed description of the procedure. In the following example, C(x) is used to represent Λ(x). <div class="mw-heading mw-heading4"><h4 id="Example_2">Example</h4>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=28" title="Edit section: Example">edit</a>]</div> Using the same data as the Peterson Gorenstein Zierler example above: <table class="wikitable"> <tbody><tr> <th>n</th> <th>Sn+1</th> <th>d</th> <th>C</th> <th>B</th> <th>b</th> <th>m </th></tr> <tr> <td>0</td> <td>732</td> <td>732</td> <td>197 x + 1</td> <td>1</td> <td>732</td> <td>1 </td></tr> <tr> <td>1</td> <td>637</td> <td>846</td> <td>173 x + 1</td> <td>1</td> <td>732</td> <td>2 </td></tr> <tr> <td>2</td> <td>762</td> <td>412</td> <td>634 x2 + 173 x + 1</td> <td>173 x + 1</td> <td>412</td> <td>1 </td></tr> <tr> <td>3</td> <td>925</td> <td>576</td> <td>329 x2 + 821 x + 1</td> <td>173 x + 1</td> <td>412</td> <td>2 </td></tr></tbody></table> The final value of C is the error locator polynomial, Λ(x). <div class="mw-heading mw-heading3"><h3 id="Euclidean_decoder">Euclidean decoder</h3>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=29" title="Edit section: Euclidean decoder">edit</a>]</div> Another iterative method for calculating both the error locator polynomial and the error value polynomial is based on Sugiyama's adaptation of the <a href="/wiki/Extended_Euclidean_algorithm" title="Extended Euclidean algorithm">extended Euclidean algorithm</a> . Define S(x), Λ(x), and Ω(x) for t syndromes and e errors: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}S(x)&=S_{t}x^{t-1}+S_{t-1}x^{t-2}+\cdots +S_{2}x+S_{1}\\[1ex]\Lambda (x)&=\Lambda _{e}x^{e}+\Lambda _{e-1}x^{e-1}+\cdots +\Lambda _{1}x+1\\[1ex]\Omega (x)&=\Omega _{e}x^{e}+\Omega _{e-1}x^{e-1}+\cdots +\Omega _{1}x+\Omega _{0}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.73em 0.73em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> <mo>+</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}S(x)&=S_{t}x^{t-1}+S_{t-1}x^{t-2}+\cdots +S_{2}x+S_{1}\\[1ex]\Lambda (x)&=\Lambda _{e}x^{e}+\Lambda _{e-1}x^{e-1}+\cdots +\Lambda _{1}x+1\\[1ex]\Omega (x)&=\Omega _{e}x^{e}+\Omega _{e-1}x^{e-1}+\cdots +\Omega _{1}x+\Omega _{0}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a84cd05dee4a72d23c59b3f95c4efdf3c8703600" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:44.154ex; height:11.843ex;" alt="{\displaystyle {\begin{aligned}S(x)&=S_{t}x^{t-1}+S_{t-1}x^{t-2}+\cdots +S_{2}x+S_{1}\\[1ex]\Lambda (x)&=\Lambda _{e}x^{e}+\Lambda _{e-1}x^{e-1}+\cdots +\Lambda _{1}x+1\\[1ex]\Omega (x)&=\Omega _{e}x^{e}+\Omega _{e-1}x^{e-1}+\cdots +\Omega _{1}x+\Omega _{0}\end{aligned}}}" /> The key equation is: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda (x)S(x)=Q(x)x^{t}+\Omega (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo>+</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda (x)S(x)=Q(x)x^{t}+\Omega (x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c956ce036d1a84f9edd391c05b086fa77c0215e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.279ex; height:3.009ex;" alt="{\displaystyle \Lambda (x)S(x)=Q(x)x^{t}+\Omega (x)}" /> For t = 6 and e = 3: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\Lambda _{3}S_{6}&x^{8}\\\Lambda _{2}S_{6}+\Lambda _{3}S_{5}&x^{7}\\\Lambda _{1}S_{6}+\Lambda _{2}S_{5}+\Lambda _{3}S_{4}&x^{6}\\S_{6}+\Lambda _{1}S_{5}+\Lambda _{2}S_{4}+\Lambda _{3}S_{3}&x^{5}\\S_{5}+\Lambda _{1}S_{4}+\Lambda _{2}S_{3}+\Lambda _{3}S_{2}&x^{4}\\S_{4}+\Lambda _{1}S_{3}+\Lambda _{2}S_{2}+\Lambda _{3}S_{1}&x^{3}\\S_{3}+\Lambda _{1}S_{2}+\Lambda _{2}S_{1}&x^{2}\\S_{2}+\Lambda _{1}S_{1}&x\\S_{1}\end{bmatrix}}={\begin{bmatrix}Q_{2}x^{8}\\Q_{1}x^{7}\\Q_{0}x^{6}\\0\\0\\0\\\Omega _{2}x^{2}\\\Omega _{1}x\\\Omega _{0}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mtd> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mtd> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\Lambda _{3}S_{6}&x^{8}\\\Lambda _{2}S_{6}+\Lambda _{3}S_{5}&x^{7}\\\Lambda _{1}S_{6}+\Lambda _{2}S_{5}+\Lambda _{3}S_{4}&x^{6}\\S_{6}+\Lambda _{1}S_{5}+\Lambda _{2}S_{4}+\Lambda _{3}S_{3}&x^{5}\\S_{5}+\Lambda _{1}S_{4}+\Lambda _{2}S_{3}+\Lambda _{3}S_{2}&x^{4}\\S_{4}+\Lambda _{1}S_{3}+\Lambda _{2}S_{2}+\Lambda _{3}S_{1}&x^{3}\\S_{3}+\Lambda _{1}S_{2}+\Lambda _{2}S_{1}&x^{2}\\S_{2}+\Lambda _{1}S_{1}&x\\S_{1}\end{bmatrix}}={\begin{bmatrix}Q_{2}x^{8}\\Q_{1}x^{7}\\Q_{0}x^{6}\\0\\0\\0\\\Omega _{2}x^{2}\\\Omega _{1}x\\\Omega _{0}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb62fd828f24b5afa5b907963ae3ab919b116957" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.171ex; width:47.225ex; height:29.509ex;" alt="{\displaystyle {\begin{bmatrix}\Lambda _{3}S_{6}&x^{8}\\\Lambda _{2}S_{6}+\Lambda _{3}S_{5}&x^{7}\\\Lambda _{1}S_{6}+\Lambda _{2}S_{5}+\Lambda _{3}S_{4}&x^{6}\\S_{6}+\Lambda _{1}S_{5}+\Lambda _{2}S_{4}+\Lambda _{3}S_{3}&x^{5}\\S_{5}+\Lambda _{1}S_{4}+\Lambda _{2}S_{3}+\Lambda _{3}S_{2}&x^{4}\\S_{4}+\Lambda _{1}S_{3}+\Lambda _{2}S_{2}+\Lambda _{3}S_{1}&x^{3}\\S_{3}+\Lambda _{1}S_{2}+\Lambda _{2}S_{1}&x^{2}\\S_{2}+\Lambda _{1}S_{1}&x\\S_{1}\end{bmatrix}}={\begin{bmatrix}Q_{2}x^{8}\\Q_{1}x^{7}\\Q_{0}x^{6}\\0\\0\\0\\\Omega _{2}x^{2}\\\Omega _{1}x\\\Omega _{0}\end{bmatrix}}}" /> The middle terms are zero due to the relationship between Λ and syndromes. The extended Euclidean algorithm can find a series of polynomials of the form <style data-mw-deduplicate="TemplateStyles:r996643573">.mw-parser-output .block-indent{padding-left:3em;padding-right:0;overflow:hidden}</style><div class="block-indent" style="padding-left: 1.5em;">Ai(x) S(x) + Bi(x) xt = Ri(x)</div> where the degree of R decreases as i increases. Once the degree of Ri(x) < t/2, then <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r996643573" /><div class="block-indent" style="padding-left: 1.5em;">Ai(x) = Λ(x)</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r996643573" /><div class="block-indent" style="padding-left: 1.5em;">Bi(x) = −Q(x)</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r996643573" /><div class="block-indent" style="padding-left: 1.5em;">Ri(x) = Ω(x).</div> B(x) and Q(x) don't need to be saved, so the algorithm becomes: <pre>R−1 := xt R0  := S(x) A−1 := 0 A0  := 1 i := 0 while degree of Ri ≥ t/2 i := i + 1 Q := Ri-2 / Ri-1 Ri := Ri-2 - Q Ri-1 Ai := Ai-2 - Q Ai-1 </pre> to set low order term of Λ(x) to 1, divide Λ(x) and Ω(x) by Ai(0): <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r996643573" /><div class="block-indent" style="padding-left: 1.5em;">Λ(x) = Ai / Ai(0)</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r996643573" /><div class="block-indent" style="padding-left: 1.5em;">Ω(x) = Ri / Ai(0)</div> Ai(0) is the constant (low order) term of Ai. <div class="mw-heading mw-heading4"><h4 id="Example_3">Example</h4>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=30" title="Edit section: Example">edit</a>]</div> Using the same data as the Peterson–Gorenstein–Zierler example above: <table class="wikitable"> <tbody><tr> <th>i </th> <th>Ri </th> <th>Ai </th></tr> <tr> <td>−1 </td> <td>001 x4 + 000 x3 + 000 x2 + 000 x + 000 </td> <td>000 </td></tr> <tr> <td>0 </td> <td>925 x3 + 762 x2 + 637 x + 732 </td> <td>001 </td></tr> <tr> <td>1 </td> <td>683 x2 + 676 x + 024 </td> <td>697 x + 396 </td></tr> <tr> <td>2 </td> <td>673 x + 596 </td> <td>608 x2 + 704 x + 544 </td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r996643573" /><div class="block-indent" style="padding-left: 1.5em;">Λ(x) = A2 / 544 = 329 x2 + 821 x + 001</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r996643573" /><div class="block-indent" style="padding-left: 1.5em;">Ω(x) = R2 / 544 = 546 x + 732</div> <div class="mw-heading mw-heading3"><h3 id="Decoder_using_discrete_Fourier_transform">Decoder using discrete Fourier transform</h3>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=31" title="Edit section: Decoder using discrete Fourier transform">edit</a>]</div> A discrete Fourier transform can be used for decoding.<a href="#cite_note-17">[16]</a> To avoid conflict with syndrome names, let c(x) = s(x) the encoded codeword. r(x) and e(x) are the same as above. Define C(x), E(x), and R(x) as the discrete Fourier transforms of c(x), e(x), and r(x). Since r(x) = c(x) + e(x), and since a discrete Fourier transform is a linear operator, R(x) = C(x) + E(x). Transform r(x) to R(x) using discrete Fourier transform. Since the calculation for a discrete Fourier transform is the same as the calculation for syndromes, t coefficients of R(x) and E(x) are the same as the syndromes: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{j}=E_{j}=S_{j}=r(\alpha ^{j})\qquad {\text{for }}1\leq j\leq t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mspace width="2em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mn>1</mn> <mo>≤</mo> <mi>j</mi> <mo>≤</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{j}=E_{j}=S_{j}=r(\alpha ^{j})\qquad {\text{for }}1\leq j\leq t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/487a6e0376f4dcf2c73d687c57c1719c9f62e311" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:39.353ex; height:3.343ex;" alt="{\displaystyle R_{j}=E_{j}=S_{j}=r(\alpha ^{j})\qquad {\text{for }}1\leq j\leq t}" /> Use <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1d63c96f59d98589d923c4f0b04222feaa7283e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle R_{1}}" /> through <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d65b678ee539ac36de96b554af181ac03b7f16a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.59ex; height:2.509ex;" alt="{\displaystyle R_{t}}" /> as syndromes (they're the same) and generate the error locator polynomial using the methods from any of the above decoders. Let v = number of errors. Generate E(x) using the known coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ac42446bcd2cbb76ec8fe2895635d328da22e26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.769ex; height:2.509ex;" alt="{\displaystyle E_{1}}" /> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e1f5f5ca535e7c0078278861ca5358b1ce80ba3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.541ex; height:2.509ex;" alt="{\displaystyle E_{t}}" />, the error locator polynomial, and these formulas <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}E_{0}&=-{\frac {1}{\Lambda _{v}}}(E_{v}+\Lambda _{1}E_{v-1}+\cdots +\Lambda _{v-1}E_{1})\\E_{j}&=-(\Lambda _{1}E_{j-1}+\Lambda _{2}E_{j-2}+\cdots +\Lambda _{v}E_{j-v})&{\text{for }}t<j<n\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>−</mo> <mi>v</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mi>t</mi> <mo><</mo> <mi>j</mi> <mo><</mo> <mi>n</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}E_{0}&=-{\frac {1}{\Lambda _{v}}}(E_{v}+\Lambda _{1}E_{v-1}+\cdots +\Lambda _{v-1}E_{1})\\E_{j}&=-(\Lambda _{1}E_{j-1}+\Lambda _{2}E_{j-2}+\cdots +\Lambda _{v}E_{j-v})&{\text{for }}t<j<n\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/570f096f29da36ca12bc6b9eaee2672859b725e7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:61.011ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}E_{0}&=-{\frac {1}{\Lambda _{v}}}(E_{v}+\Lambda _{1}E_{v-1}+\cdots +\Lambda _{v-1}E_{1})\\E_{j}&=-(\Lambda _{1}E_{j-1}+\Lambda _{2}E_{j-2}+\cdots +\Lambda _{v}E_{j-v})&{\text{for }}t<j<n\end{aligned}}}" /> Then calculate C(x) = R(x) − E(x) and take the inverse transform (polynomial interpolation) of C(x) to produce c(x). <div class="mw-heading mw-heading3"><h3 id="Decoding_beyond_the_error-correction_bound">Decoding beyond the error-correction bound</h3>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=32" title="Edit section: Decoding beyond the error-correction bound">edit</a>]</div> The <a href="/wiki/Singleton_bound" title="Singleton bound">Singleton bound</a> states that the minimum distance d of a linear block code of size (n,k) is upper-bounded by n - k + 1. The distance d was usually understood to limit the error-correction capability to ⌊(d - 1) / 2⌋. The Reed–Solomon code achieves this bound with equality, and can thus correct up to ⌊(n - k) / 2⌋ errors. However, this error-correction bound is not exact. In 1999, <a href="/wiki/Madhu_Sudan" title="Madhu Sudan">Madhu Sudan</a> and <a href="/wiki/Venkatesan_Guruswami" title="Venkatesan Guruswami">Venkatesan Guruswami</a> at MIT published "Improved Decoding of Reed–Solomon and Algebraic-Geometry Codes" introducing an algorithm that allowed for the correction of errors beyond half the minimum distance of the code.<a href="#cite_note-18">[17]</a> It applies to Reed–Solomon codes and more generally to <a href="/wiki/Algebraic_geometric_code" class="mw-redirect" title="Algebraic geometric code">algebraic geometric codes</a>. This algorithm produces a list of codewords (it is a <a href="/wiki/List-decoding" class="mw-redirect" title="List-decoding">list-decoding</a> algorithm) and is based on interpolation and factorization of polynomials over GF(2m) and its extensions. In 2023, building on three exciting[<a href="/wiki/Wikipedia:Manual_of_Style/Words_to_watch#Unsupported_attributions" title="Wikipedia:Manual of Style/Words to watch">according to whom?</a>] works,<a href="#cite_note-19">[18]</a><a href="#cite_note-20">[19]</a><a href="#cite_note-21">[20]</a> coding theorists showed that Reed-Solomon codes defined over random evaluation points can actually achieve <a href="/wiki/List_decoding" title="List decoding">list decoding</a> capacity (up to n - k errors) over linear size alphabets with high probability. However, this result is combinatorial rather than algorithmic.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <div class="mw-heading mw-heading3"><h3 id="Soft-decoding">Soft-decoding</h3>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=33" title="Edit section: Soft-decoding">edit</a>]</div> The algebraic decoding methods described above are hard-decision methods, which means that for every symbol a hard decision is made about its value. For example, a decoder could associate with each symbol an additional value corresponding to the channel <a href="/wiki/Demodulator" class="mw-redirect" title="Demodulator">demodulator</a>'s confidence in the correctness of the symbol. The advent of <a href="/wiki/Low-density_parity-check_code" title="Low-density parity-check code">LDPC</a> and <a href="/wiki/Turbo_code" title="Turbo code">turbo codes</a>, which employ iterated <a href="/wiki/Soft-decision_decoding" class="mw-redirect" title="Soft-decision decoding">soft-decision</a> belief propagation decoding methods to achieve error-correction performance close to the <a href="/wiki/Shannon_limit" class="mw-redirect" title="Shannon limit">theoretical limit</a>, has spurred interest in applying soft-decision decoding to conventional algebraic codes. In 2003, Ralf Koetter and <a href="/wiki/Alexander_Vardy" title="Alexander Vardy">Alexander Vardy</a> presented a polynomial-time soft-decision algebraic list-decoding algorithm for Reed–Solomon codes, which was based upon the work by Sudan and Guruswami.<a href="#cite_note-22">[21]</a> In 2016, Steven J. Franke and Joseph H. Taylor published a novel soft-decision decoder.<a href="#cite_note-23">[22]</a> <div class="mw-heading mw-heading3"><h3 id="MATLAB_example">MATLAB example</h3>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=34" title="Edit section: MATLAB example">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Encoder">Encoder</h4>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=35" title="Edit section: Encoder">edit</a>]</div> Here we present a simple <a href="/wiki/MATLAB" title="MATLAB">MATLAB</a> implementation for an encoder. <div class="mw-highlight mw-highlight-lang-matlab mw-content-ltr mw-highlight-lines" dir="ltr"><pre>function encoded = rsEncoder(msg, m, prim_poly, n, k) % RSENCODER Encode message with the Reed-Solomon algorithm % m is the number of bits per symbol % prim_poly: Primitive polynomial p(x). Ie for DM is 301 % k is the size of the message % n is the total size (k+redundant) % Example: msg = uint8('Test') % enc_msg = rsEncoder(msg, 8, 301, 12, numel(msg)); % Get the alpha alpha = gf(2, m, prim_poly); % Get the Reed-Solomon generating polynomial g(x) g_x = genpoly(k, n, alpha); % Multiply the information by X^(n-k), or just pad with zeros at the end to % get space to add the redundant information msg_padded = gf([msg zeros(1, n - k)], m, prim_poly); % Get the remainder of the division of the extended message by the % Reed-Solomon generating polynomial g(x) [~, remainder] = deconv(msg_padded, g_x); % Now return the message with the redundant information encoded = msg_padded - remainder; end % Find the Reed-Solomon generating polynomial g(x), by the way this is the % same as the rsgenpoly function on matlab function g = genpoly(k, n, alpha) g = 1; % A multiplication on the galois field is just a convolution for k = mod(1 : n - k, n) g = conv(g, [1 alpha .^ (k)]); end end </pre></div> <div class="mw-heading mw-heading4"><h4 id="Decoder">Decoder</h4>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=36" title="Edit section: Decoder">edit</a>]</div> Now the decoding part: <div class="mw-highlight mw-highlight-lang-matlab mw-content-ltr mw-highlight-lines" dir="ltr"><pre>function [decoded, error_pos, error_mag, g, S] = rsDecoder(encoded, m, prim_poly, n, k) % RSDECODER Decode a Reed-Solomon encoded message % Example: % [dec, ~, ~, ~, ~] = rsDecoder(enc_msg, 8, 301, 12, numel(msg)) max_errors = floor((n - k) / 2); orig_vals = encoded.x; % Initialize the error vector errors = zeros(1, n); g = []; S = []; % Get the alpha alpha = gf(2, m, prim_poly); % Find the syndromes (Check if dividing the message by the generator % polynomial the result is zero) Synd = polyval(encoded, alpha .^ (1:n - k)); Syndromes = trim(Synd); % If all syndromes are zeros (perfectly divisible) there are no errors if isempty(Syndromes.x) decoded = orig_vals(1:k); error_pos = []; error_mag = []; g = []; S = Synd; return; end % Prepare for the euclidean algorithm (Used to find the error locating % polynomials) r0 = [1, zeros(1, 2 * max_errors)]; r0 = gf(r0, m, prim_poly); r0 = trim(r0); size_r0 = length(r0); r1 = Syndromes; f0 = gf([zeros(1, size_r0 - 1) 1], m, prim_poly); f1 = gf(zeros(1, size_r0), m, prim_poly); g0 = f1; g1 = f0; % Do the euclidean algorithm on the polynomials r0(x) and Syndromes(x) in % order to find the error locating polynomial while true % Do a long division [quotient, remainder] = deconv(r0, r1); % Add some zeros quotient = pad(quotient, length(g1)); % Find quotient*g1 and pad c = conv(quotient, g1); c = trim(c); c = pad(c, length(g0)); % Update g as g0-quotient*g1 g = g0 - c; % Check if the degree of remainder(x) is less than max_errors if all(remainder(1:end - max_errors) == 0) break; end % Update r0, r1, g0, g1 and remove leading zeros r0 = trim(r1); r1 = trim(remainder); g0 = g1; g1 = g; end % Remove leading zeros g = trim(g); % Find the zeros of the error polynomial on this galois field evalPoly = polyval(g, alpha .^ (n - 1 : - 1 : 0)); error_pos = gf(find(evalPoly == 0), m); % If no error position is found we return the received work, because % basically is nothing that we could do and we return the received message if isempty(error_pos) decoded = orig_vals(1:k); error_mag = []; return; end % Prepare a linear system to solve the error polynomial and find the error % magnitudes size_error = length(error_pos); Syndrome_Vals = Syndromes.x; b(:, 1) = Syndrome_Vals(1:size_error); for idx = 1 : size_error e = alpha .^ (idx * (n - error_pos.x)); err = e.x; er(idx, :) = err; end % Solve the linear system error_mag = (gf(er, m, prim_poly) \ gf(b, m, prim_poly))'; % Put the error magnitude on the error vector errors(error_pos.x) = error_mag.x; % Bring this vector to the galois field errors_gf = gf(errors, m, prim_poly); % Now to fix the errors just add with the encoded code decoded_gf = encoded(1:k) + errors_gf(1:k); decoded = decoded_gf.x; end % Remove leading zeros from Galois array function gt = trim(g) gx = g.x; gt = gf(gx(find(gx, 1) : end), g.m, g.prim_poly); end % Add leading zeros function xpad = pad(x, k) len = length(x); if len < k xpad = [zeros(1, k - len) x]; end end </pre></div> <div class="mw-heading mw-heading2"><h2 id="Reed_Solomon_original_view_decoders">Reed Solomon original view decoders</h2>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=37" title="Edit section: Reed Solomon original view decoders">edit</a>]</div> The decoders described in this section use the Reed Solomon original view of a codeword as a sequence of polynomial values where the polynomial is based on the message to be encoded. The same set of fixed values are used by the encoder and decoder, and the decoder recovers the encoding polynomial (and optionally an error locating polynomial) from the received message. <div class="mw-heading mw-heading3"><h3 id="Theoretical_decoder">Theoretical decoder</h3>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=38" title="Edit section: Theoretical decoder">edit</a>]</div> Reed and Solomon described a theoretical decoder that corrected errors by finding the most popular message polynomial.<a href="#cite_note-ReedSolomon-1">[1]</a> The decoder only knows the set of values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbf42ecda092975c9c69dae84e16182ba5fe2e07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{1}}" /> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}" /> and which encoding method was used to generate the codeword's sequence of values. The original message, the polynomial, and any errors are unknown. A decoding procedure could use a method like Lagrange interpolation on various subsets of n codeword values taken k at a time to repeatedly produce potential polynomials, until a sufficient number of matching polynomials are produced to reasonably eliminate any errors in the received codeword. Once a polynomial is determined, then any errors in the codeword can be corrected, by recalculating the corresponding codeword values. Unfortunately, in all but the simplest of cases, there are too many subsets, so the algorithm is impractical. The number of subsets is the <a href="/wiki/Binomial_coefficient" title="Binomial coefficient">binomial coefficient</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\binom {n}{k}}={n! \over (n-k)!k!}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\binom {n}{k}}={n! \over (n-k)!k!}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a36e5f3655c64191556de4752708dedcf6bc830e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.222ex; height:4.343ex;" alt="{\textstyle {\binom {n}{k}}={n! \over (n-k)!k!}}" />, and the number of subsets is infeasible for even modest codes. For a (255,249) code that can correct 3 errors, the naïve theoretical decoder would examine 359 billion subsets.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <div class="mw-heading mw-heading3"><h3 id="Berlekamp_Welch_decoder">Berlekamp Welch decoder</h3>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=39" title="Edit section: Berlekamp Welch decoder">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></div></td><td class="mbox-text"><div class="mbox-text-span">This section does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a>. Please help <a href="/wiki/Special:EditPage/Reed%E2%80%93Solomon_error_correction" title="Special:EditPage/Reed–Solomon error correction">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>. (March 2025) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> In 1986, a decoder known as the <a href="/wiki/Berlekamp%E2%80%93Welch_algorithm" title="Berlekamp–Welch algorithm">Berlekamp–Welch algorithm</a> was developed as a decoder that is able to recover the original message polynomial as well as an error "locator" polynomial that produces zeroes for the input values that correspond to errors, with time complexity O(n3), where n is the number of values in a message. The recovered polynomial is then used to recover (recalculate as needed) the original message. <div class="mw-heading mw-heading4"><h4 id="Example_4">Example</h4>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=40" title="Edit section: Example">edit</a>]</div> Using RS(7,3), GF(929), and the set of evaluation points ai = i − 1 <dl><dd> a = {0, 1, 2, 3, 4, 5, 6} </dd></dl> If the message polynomial is <dl><dd> p(x) = 003x2 + 002x + 001</dd></dl> The codeword is <dl><dd> c = {001, 006, 017, 034, 057, 086, 121} </dd></dl> Errors in transmission might cause this to be received instead. <dl><dd> b = c + e = {001, 006, 123, 456, 057, 086, 121} </dd></dl> The key equation is: <dl><dd>biE(ai) - Q(ai) = 0</dd></dl> Assume maximum number of errors: e = 2. The key equation becomes: <dl><dd>bi(e0 + e1ai) - (q0 + q1ai + q2 a2 i + q3a3 i + q4a4 i) = - bia2 i</dd></dl> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}001&000&928&000&000&000&000\\006&006&928&928&928&928&928\\123&246&928&927&925&921&913\\456&439&928&926&920&902&848\\057&228&928&925&913&865&673\\086&430&928&924&904&804&304\\121&726&928&923&893&713&562\end{bmatrix}}{\begin{bmatrix}e_{0}\\e_{1}\\q_{0}\\q_{1}\\q_{2}\\q_{3}\\q_{4}\end{bmatrix}}={\begin{bmatrix}000\\923\\437\\541\\017\\637\\289\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>001</mn> </mtd> <mtd> <mn>000</mn> </mtd> <mtd> <mn>928</mn> </mtd> <mtd> <mn>000</mn> </mtd> <mtd> <mn>000</mn> </mtd> <mtd> <mn>000</mn> </mtd> <mtd> <mn>000</mn> </mtd> </mtr> <mtr> <mtd> <mn>006</mn> </mtd> <mtd> <mn>006</mn> </mtd> <mtd> <mn>928</mn> </mtd> <mtd> <mn>928</mn> </mtd> <mtd> <mn>928</mn> </mtd> <mtd> <mn>928</mn> </mtd> <mtd> <mn>928</mn> </mtd> </mtr> <mtr> <mtd> <mn>123</mn> </mtd> <mtd> <mn>246</mn> </mtd> <mtd> <mn>928</mn> </mtd> <mtd> <mn>927</mn> </mtd> <mtd> <mn>925</mn> </mtd> <mtd> <mn>921</mn> </mtd> <mtd> <mn>913</mn> </mtd> </mtr> <mtr> <mtd> <mn>456</mn> </mtd> <mtd> <mn>439</mn> </mtd> <mtd> <mn>928</mn> </mtd> <mtd> <mn>926</mn> </mtd> <mtd> <mn>920</mn> </mtd> <mtd> <mn>902</mn> </mtd> <mtd> <mn>848</mn> </mtd> </mtr> <mtr> <mtd> <mn>057</mn> </mtd> <mtd> <mn>228</mn> </mtd> <mtd> <mn>928</mn> </mtd> <mtd> <mn>925</mn> </mtd> <mtd> <mn>913</mn> </mtd> <mtd> <mn>865</mn> </mtd> <mtd> <mn>673</mn> </mtd> </mtr> <mtr> <mtd> <mn>086</mn> </mtd> <mtd> <mn>430</mn> </mtd> <mtd> <mn>928</mn> </mtd> <mtd> <mn>924</mn> </mtd> <mtd> <mn>904</mn> </mtd> <mtd> <mn>804</mn> </mtd> <mtd> <mn>304</mn> </mtd> </mtr> <mtr> <mtd> <mn>121</mn> </mtd> <mtd> <mn>726</mn> </mtd> <mtd> <mn>928</mn> </mtd> <mtd> <mn>923</mn> </mtd> <mtd> <mn>893</mn> </mtd> <mtd> <mn>713</mn> </mtd> <mtd> <mn>562</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>000</mn> </mtd> </mtr> <mtr> <mtd> <mn>923</mn> </mtd> </mtr> <mtr> <mtd> <mn>437</mn> </mtd> </mtr> <mtr> <mtd> <mn>541</mn> </mtd> </mtr> <mtr> <mtd> <mn>017</mn> </mtd> </mtr> <mtr> <mtd> <mn>637</mn> </mtd> </mtr> <mtr> <mtd> <mn>289</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}001&000&928&000&000&000&000\\006&006&928&928&928&928&928\\123&246&928&927&925&921&913\\456&439&928&926&920&902&848\\057&228&928&925&913&865&673\\086&430&928&924&904&804&304\\121&726&928&923&893&713&562\end{bmatrix}}{\begin{bmatrix}e_{0}\\e_{1}\\q_{0}\\q_{1}\\q_{2}\\q_{3}\\q_{4}\end{bmatrix}}={\begin{bmatrix}000\\923\\437\\541\\017\\637\\289\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7a266b6f5d5561dc5563555f6b641878dae8942" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.671ex; width:58.627ex; height:22.343ex;" alt="{\displaystyle {\begin{bmatrix}001&000&928&000&000&000&000\\006&006&928&928&928&928&928\\123&246&928&927&925&921&913\\456&439&928&926&920&902&848\\057&228&928&925&913&865&673\\086&430&928&924&904&804&304\\121&726&928&923&893&713&562\end{bmatrix}}{\begin{bmatrix}e_{0}\\e_{1}\\q_{0}\\q_{1}\\q_{2}\\q_{3}\\q_{4}\end{bmatrix}}={\begin{bmatrix}000\\923\\437\\541\\017\\637\\289\end{bmatrix}}}" /> Using <a href="/wiki/Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}001&000&000&000&000&000&000\\000&001&000&000&000&000&000\\000&000&001&000&000&000&000\\000&000&000&001&000&000&000\\000&000&000&000&001&000&000\\000&000&000&000&000&001&000\\000&000&000&000&000&000&001\end{bmatrix}}{\begin{bmatrix}e_{0}\\e_{1}\\q_{0}\\q_{1}\\q_{2}\\q_{3}\\q_{4}\end{bmatrix}}={\begin{bmatrix}006\\924\\006\\007\\009\\916\\003\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>001</mn> </mtd> <mtd> <mn>000</mn> </mtd> <mtd> <mn>000</mn> </mtd> <mtd> <mn>000</mn> </mtd> <mtd> <mn>000</mn> </mtd> <mtd> <mn>000</mn> </mtd> <mtd> <mn>000</mn> </mtd> </mtr> <mtr> <mtd> <mn>000</mn> </mtd> <mtd> <mn>001</mn> </mtd> <mtd> <mn>000</mn> </mtd> <mtd> <mn>000</mn> </mtd> <mtd> <mn>000</mn> </mtd> <mtd> <mn>000</mn> </mtd> <mtd> <mn>000</mn> </mtd> </mtr> <mtr> <mtd> <mn>000</mn> </mtd> <mtd> <mn>000</mn> </mtd> <mtd> <mn>001</mn> </mtd> <mtd> <mn>000</mn> </mtd> <mtd> <mn>000</mn> </mtd> <mtd> <mn>000</mn> </mtd> <mtd> <mn>000</mn> </mtd> </mtr> <mtr> <mtd> <mn>000</mn> </mtd> <mtd> <mn>000</mn> </mtd> <mtd> <mn>000</mn> </mtd> <mtd> <mn>001</mn> </mtd> <mtd> <mn>000</mn> </mtd> <mtd> <mn>000</mn> </mtd> <mtd> <mn>000</mn> </mtd> </mtr> <mtr> <mtd> <mn>000</mn> </mtd> <mtd> <mn>000</mn> </mtd> <mtd> <mn>000</mn> </mtd> <mtd> <mn>000</mn> </mtd> <mtd> <mn>001</mn> </mtd> <mtd> <mn>000</mn> </mtd> <mtd> <mn>000</mn> </mtd> </mtr> <mtr> <mtd> <mn>000</mn> </mtd> <mtd> <mn>000</mn> </mtd> <mtd> <mn>000</mn> </mtd> <mtd> <mn>000</mn> </mtd> <mtd> <mn>000</mn> </mtd> <mtd> <mn>001</mn> </mtd> <mtd> <mn>000</mn> </mtd> </mtr> <mtr> <mtd> <mn>000</mn> </mtd> <mtd> <mn>000</mn> </mtd> <mtd> <mn>000</mn> </mtd> <mtd> <mn>000</mn> </mtd> <mtd> <mn>000</mn> </mtd> <mtd> <mn>000</mn> </mtd> <mtd> <mn>001</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>006</mn> </mtd> </mtr> <mtr> <mtd> <mn>924</mn> </mtd> </mtr> <mtr> <mtd> <mn>006</mn> </mtd> </mtr> <mtr> <mtd> <mn>007</mn> </mtd> </mtr> <mtr> <mtd> <mn>009</mn> </mtd> </mtr> <mtr> <mtd> <mn>916</mn> </mtd> </mtr> <mtr> <mtd> <mn>003</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}001&000&000&000&000&000&000\\000&001&000&000&000&000&000\\000&000&001&000&000&000&000\\000&000&000&001&000&000&000\\000&000&000&000&001&000&000\\000&000&000&000&000&001&000\\000&000&000&000&000&000&001\end{bmatrix}}{\begin{bmatrix}e_{0}\\e_{1}\\q_{0}\\q_{1}\\q_{2}\\q_{3}\\q_{4}\end{bmatrix}}={\begin{bmatrix}006\\924\\006\\007\\009\\916\\003\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/977104b3920fcbdfe629371a76ae92953c978886" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.671ex; width:58.627ex; height:22.343ex;" alt="{\displaystyle {\begin{bmatrix}001&000&000&000&000&000&000\\000&001&000&000&000&000&000\\000&000&001&000&000&000&000\\000&000&000&001&000&000&000\\000&000&000&000&001&000&000\\000&000&000&000&000&001&000\\000&000&000&000&000&000&001\end{bmatrix}}{\begin{bmatrix}e_{0}\\e_{1}\\q_{0}\\q_{1}\\q_{2}\\q_{3}\\q_{4}\end{bmatrix}}={\begin{bmatrix}006\\924\\006\\007\\009\\916\\003\end{bmatrix}}}" /> <dl><dd> Q(x) = 003x4 + 916x3 + 009x2 + 007x + 006</dd> <dd> E(x) = 001x2 + 924x + 006</dd></dl> <dl><dd> <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠Q(x)/E(x)⁠ = P(x) = 003x2 + 002x + 001</dd></dl> Recalculate P(x) where E(x) = 0 : {2, 3} to correct b resulting in the corrected codeword: <dl><dd> c = {001, 006, 017, 034, 057, 086, 121} }}</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Gao_decoder">Gao decoder</h3>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=41" title="Edit section: Gao decoder">edit</a>]</div> In 2002, an improved decoder was developed by Shuhong Gao, based on the extended Euclid algorithm.<a href="#cite_note-gao-6">[6]</a> <div class="mw-heading mw-heading4"><h4 id="Example_5">Example</h4>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=42" title="Edit section: Example">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444" /><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></div></td><td class="mbox-text"><div class="mbox-text-span">This section does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a>. Please help <a href="/wiki/Special:EditPage/Reed%E2%80%93Solomon_error_correction" title="Special:EditPage/Reed–Solomon error correction">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>. (March 2025) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> Using the same data as the Berlekamp Welch example above: <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{-1}=\prod _{i=1}^{n}(x-a_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{-1}=\prod _{i=1}^{n}(x-a_{i})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c5ead155b26eb5317b0b981762b50003615c6d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.174ex; height:6.843ex;" alt="{\displaystyle R_{-1}=\prod _{i=1}^{n}(x-a_{i})}" /></li> <li> R0 = Lagrange interpolation of {ai, b(ai)} for i = 1 to n</li> <li> A-1 = 0</li> <li> A0 = 1</li></ul> <table class="wikitable"> <tbody><tr> <th>i </th> <th>Ri </th> <th>Ai </th></tr> <tr> <td>−1 </td> <td>001x7 + 908x6 + 175x5 + 194x4 + 695x3 + 094x2 + 720x + 000 </td> <td>000 </td></tr> <tr> <td>0 </td> <td>055x6 + 440x5 + 497x4 + 904x3 + 424x2 + 472x + 001 </td> <td>001 </td></tr> <tr> <td>1 </td> <td>702x5 + 845x4 + 691x3 + 461x2 + 327x + 237 </td> <td>152 x + 237 </td></tr> <tr> <td>2 </td> <td>266x4 + 086x3 + 798x2 + 311x + 532 </td> <td>708x2 + 176x + 532 </td></tr></tbody></table> <dl><dd> Q(x) = R2 = 266x4 + 086x3 + 798x2 + 311x + 532</dd></dl> <dl><dd> E(x) = A2 = 708x2 + 176x + 532</dd></dl> divide Q(x) and E(x) by most significant coefficient of E(x) = 708. (Optional)[<a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify">further explanation needed</a>] <dl><dd> Q(x) = 003x4 + 916x3 + 009x2 + 007x + 006</dd> <dd> E(x) = 001x2 + 924x + 006</dd></dl> <dl><dd> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" />⁠Q(x)/E(x)⁠ = P(x) = 003x2 + 002x + 001</dd></dl> Recalculate P(x) where E(x) = 0 : {2, 3} to correct b resulting in the corrected codeword: <dl><dd> c = {001, 006, 017, 034, 057, 086, 121} </dd></dl> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=43" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/BCH_code" title="BCH code">BCH code</a></li> <li><a href="/wiki/Berlekamp%E2%80%93Massey_algorithm" title="Berlekamp–Massey algorithm">Berlekamp–Massey algorithm</a></li> <li><a href="/wiki/Berlekamp%E2%80%93Welch_algorithm" title="Berlekamp–Welch algorithm">Berlekamp–Welch algorithm</a></li> <li><a href="/wiki/Chien_search" title="Chien search">Chien search</a></li> <li><a href="/wiki/Cyclic_code" title="Cyclic code">Cyclic code</a></li> <li><a href="/wiki/Folded_Reed%E2%80%93Solomon_code" title="Folded Reed–Solomon code">Folded Reed–Solomon code</a></li> <li><a href="/wiki/Forward_error_correction" class="mw-redirect" title="Forward error correction">Forward error correction</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=44" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-10"><a href="#cite_ref-10">^</a> Authors in Andrews et al. (2007), provide simulation results which show that for the same code rate (1/6) turbo codes outperform Reed-Solomon concatenated codes up to 2 dB (<a href="/wiki/Bit_error_rate" title="Bit error rate">bit error rate</a>).<a href="#cite_note-Andrews,_2007-9">[9]</a> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=45" title="Edit section: References">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626" /><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-ReedSolomon-1">^ <a href="#cite_ref-ReedSolomon_1-0">a</a> <a href="#cite_ref-ReedSolomon_1-1">b</a> <a href="#cite_ref-ReedSolomon_1-2">c</a> <a href="#cite_ref-ReedSolomon_1-3">d</a> <a href="#cite_ref-ReedSolomon_1-4">e</a> <a href="#cite_ref-ReedSolomon_1-5">f</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFReedSolomon1960" class="citation journal cs1"><a href="/wiki/Irving_S._Reed" title="Irving S. Reed">Reed, Irving S.</a>; <a href="/wiki/Gustave_Solomon" title="Gustave Solomon">Solomon, Gustave</a> (1960). <a rel="nofollow" class="external text" href="https://sites.math.rutgers.edu/~zeilberg/akherim/ReedS1960.pdf">"Polynomial Codes over Certain Finite Fields"</a> (PDF). Journal of the Society for Industrial and Applied Mathematics. 8 (2): 300–304. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F0108018">10.1137/0108018</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Society+for+Industrial+and+Applied+Mathematics&rft.atitle=Polynomial+Codes+over+Certain+Finite+Fields&rft.volume=8&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E300-%3C%2Fspan%3E304&rft.date=1960&rft_id=info%3Adoi%2F10.1137%2F0108018&rft.aulast=Reed&rft.aufirst=Irving+S.&rft.au=Solomon%2C+Gustave&rft_id=https%3A%2F%2Fsites.math.rutgers.edu%2F~zeilberg%2Fakherim%2FReedS1960.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReed%E2%80%93Solomon+error+correction" class="Z3988"> </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGorensteinZierler1961" class="citation journal cs1">Gorenstein, D.; Zierler, N. (June 1961). "A class of cyclic linear error-correcting codes in pm symbols". J. SIAM. 9 (2): 207–214. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F0109020">10.1137/0109020</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2098821">2098821</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+SIAM&rft.atitle=A+class+of+cyclic+linear+error-correcting+codes+in+p%3Csup%3Em%3C%2Fsup%3E+symbols&rft.volume=9&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E207-%3C%2Fspan%3E214&rft.date=1961-06&rft_id=info%3Adoi%2F10.1137%2F0109020&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2098821%23id-name%3DJSTOR&rft.aulast=Gorenstein&rft.aufirst=D.&rft.au=Zierler%2C+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReed%E2%80%93Solomon+error+correction" class="Z3988"> </li> <li id="cite_note-Peterson61-3">^ <a href="#cite_ref-Peterson61_3-0">a</a> <a href="#cite_ref-Peterson61_3-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPeterson1961" class="citation book cs1">Peterson, W. Wesley (1961). Error-Correcting Codes. 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MIT Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-585-30709-1" title="Special:BookSources/978-0-585-30709-1"><bdi>978-0-585-30709-1</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/45727875">45727875</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Error+Correcting+Codes&rft.edition=2nd&rft.pub=MIT+Press&rft.date=1996&rft_id=info%3Aoclcnum%2F45727875&rft.isbn=978-0-585-30709-1&rft.aulast=Peterson&rft.aufirst=W.+Wesley&rft.au=Weldon%2C+E.+J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D5kfwlFeklx0C%26pg%3DPP11%26dq%3D%25229780262160391%2522&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReed%E2%80%93Solomon+error+correction" class="Z3988"> </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSugiyamaKasaharaHirasawaNamekawa1975" class="citation journal cs1">Sugiyama, Y.; Kasahara, M.; Hirasawa, S.; Namekawa, T. 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(eds.), Reed–Solomon Codes and Their Applications, <a href="/wiki/IEEE_Press" class="mw-redirect" title="IEEE Press">IEEE Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7803-1025-4" title="Special:BookSources/978-0-7803-1025-4"><bdi>978-0-7803-1025-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Reed%E2%80%93Solomon+Codes+and+the+Compact+Disc&rft.btitle=Reed%E2%80%93Solomon+Codes+and+Their+Applications&rft.pub=IEEE+Press&rft.date=1994&rft.isbn=978-0-7803-1025-4&rft.aulast=Immink&rft.aufirst=K.+A.+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReed%E2%80%93Solomon+error+correction" class="Z3988"> </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHagenauerOfferPapke1994" class="citation book cs1">Hagenauer, J.; Offer, E.; Papke, L. 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FOCS 2023, Santa Cruz, CA, USA, 2023. pp. 164–176. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2304.01403">2304.01403</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FFOCS57990.2023.00019">10.1109/FOCS57990.2023.00019</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/979-8-3503-1894-4" title="Special:BookSources/979-8-3503-1894-4"><bdi>979-8-3503-1894-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Randomly+Punctured+Reed-Solomon+Codes+Achieve+the+List+Decoding+Capacity+over+Polynomial-Size+Alphabets&rft.btitle=2023+IEEE+64th+Annual+Symposium+on+Foundations+of+Computer+Science+%28FOCS%29&rft.series=FOCS+2023%2C+Santa+Cruz%2C+CA%2C+USA%2C+2023&rft.pages=%3Cspan+class%3D%22nowrap%22%3E164-%3C%2Fspan%3E176&rft.date=2023&rft_id=info%3Aarxiv%2F2304.01403&rft_id=info%3Adoi%2F10.1109%2FFOCS57990.2023.00019&rft.isbn=979-8-3503-1894-4&rft.aulast=Guo&rft.aufirst=Zeyu&rft.au=Zhang%2C+Zihan&rft_id=https%3A%2F%2Fdoi.org%2F10.1109%2FFOCS57990.2023.00019&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReed%E2%80%93Solomon+error+correction" class="Z3988"> </li> <li id="cite_note-21"><a href="#cite_ref-21">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAlrabiahGuruswamiLi2023" class="citation cs2">Alrabiah, Omar; Guruswami, Venkatesan; Li, Ray (2023-08-18), Randomly punctured Reed--Solomon codes achieve list-decoding capacity over linear-sized fields, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2304.09445">2304.09445</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Randomly+punctured+Reed--Solomon+codes+achieve+list-decoding+capacity+over+linear-sized+fields&rft.date=2023-08-18&rft_id=info%3Aarxiv%2F2304.09445&rft.aulast=Alrabiah&rft.aufirst=Omar&rft.au=Guruswami%2C+Venkatesan&rft.au=Li%2C+Ray&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReed%E2%80%93Solomon+error+correction" class="Z3988"> </li> <li id="cite_note-22"><a href="#cite_ref-22">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKoetterVardy2003" class="citation journal cs1">Koetter, Ralf; Vardy, Alexander (2003). "Algebraic soft-decision decoding of Reed–Solomon codes". <a href="/wiki/IEEE_Transactions_on_Information_Theory" title="IEEE Transactions on Information Theory">IEEE Transactions on Information Theory</a>. 49 (11): 2809–2825. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.13.2021">10.1.1.13.2021</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTIT.2003.819332">10.1109/TIT.2003.819332</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Transactions+on+Information+Theory&rft.atitle=Algebraic+soft-decision+decoding+of+Reed%E2%80%93Solomon+codes&rft.volume=49&rft.issue=11&rft.pages=%3Cspan+class%3D%22nowrap%22%3E2809-%3C%2Fspan%3E2825&rft.date=2003&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.13.2021%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1109%2FTIT.2003.819332&rft.aulast=Koetter&rft.aufirst=Ralf&rft.au=Vardy%2C+Alexander&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReed%E2%80%93Solomon+error+correction" class="Z3988"> </li> <li id="cite_note-23"><a href="#cite_ref-23">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFFrankeTaylor2016" class="citation journal cs1">Franke, Steven J.; Taylor, Joseph H. (2016). <a rel="nofollow" class="external text" href="http://physics.princeton.edu/pulsar/K1JT/FrankeTaylor_QEX_2016.pdf">"Open Source Soft-Decision Decoder for the JT65 (63,12) Reed–Solomon Code"</a> (PDF). QEX (May/June): 8–17. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170309134627/http://www.physics.princeton.edu/pulsar/k1jt/FrankeTaylor_QEX_2016.pdf">Archived</a> (PDF) from the original on 2017-03-09. Retrieved 2017-06-07.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=QEX&rft.atitle=Open+Source+Soft-Decision+Decoder+for+the+JT65+%2863%2C12%29+Reed%E2%80%93Solomon+Code&rft.issue=May%2FJune&rft.pages=%3Cspan+class%3D%22nowrap%22%3E8-%3C%2Fspan%3E17&rft.date=2016&rft.aulast=Franke&rft.aufirst=Steven+J.&rft.au=Taylor%2C+Joseph+H.&rft_id=http%3A%2F%2Fphysics.princeton.edu%2Fpulsar%2FK1JT%2FFrankeTaylor_QEX_2016.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReed%E2%80%93Solomon+error+correction" class="Z3988"> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=46" title="Edit section: External links">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Information_and_tutorials">Information and tutorials</h3>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=47" title="Edit section: Information and tutorials">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWelch1997" class="citation cs2">Welch, L. R. (1997), <a rel="nofollow" class="external text" href="http://csi.usc.edu/PDF/RSoriginal.pdf">The Original View of Reed–Solomon Codes</a> (PDF), Lecture Notes</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Original+View+of+Reed%E2%80%93Solomon+Codes&rft.series=Lecture+Notes&rft.date=1997&rft.aulast=Welch&rft.aufirst=L.+R.&rft_id=http%3A%2F%2Fcsi.usc.edu%2FPDF%2FRSoriginal.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReed%E2%80%93Solomon+error+correction" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKoetter2005" class="citation cs2">Koetter, Ralf (2005), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130313033107/http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-451-principles-of-digital-communication-ii-spring-2005/lecture-notes/lecture-10-reed-solomon-codes/">Reed–Solomon Codes</a>, MIT Lecture Notes 6.451 (Video), archived from <a rel="nofollow" class="external text" href="http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-451-principles-of-digital-communication-ii-spring-2005/lecture-notes/lecture-10-reed-solomon-codes/">the original</a> on 2013-03-13</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Reed%E2%80%93Solomon+Codes&rft.series=MIT+Lecture+Notes+6.451+%28Video%29&rft.date=2005&rft.aulast=Koetter&rft.aufirst=Ralf&rft_id=http%3A%2F%2Focw.mit.edu%2Fcourses%2Felectrical-engineering-and-computer-science%2F6-451-principles-of-digital-communication-ii-spring-2005%2Flecture-notes%2Flecture-10-reed-solomon-codes%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReed%E2%80%93Solomon+error+correction" class="Z3988"></li></ul> <ul><li><a rel="nofollow" class="external text" href="https://www.cs.cmu.edu/~guyb/realworld/reedsolomon/reed_solomon_codes.html">Introduction to Reed–Solomon codes: principles, architecture and implementation</a> (CMU)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGeisel1990" class="citation cs2">Geisel, William A. (August 1990), <a rel="nofollow" class="external text" href="https://ntrs.nasa.gov/citations/19900019023">Tutorial on Reed–Solomon Error Correction Coding</a>, Technical Memorandum, <a href="/wiki/NASA" title="NASA">NASA</a>, TM-102162</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Tutorial+on+Reed%E2%80%93Solomon+Error+Correction+Coding&rft.series=Technical+Memorandum&rft.pub=NASA&rft.date=1990-08&rft.aulast=Geisel&rft.aufirst=William+A.&rft_id=https%3A%2F%2Fntrs.nasa.gov%2Fcitations%2F19900019023&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReed%E2%80%93Solomon+error+correction" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFReid1995" class="citation cs2">Reid, Jeff A. (April 1995), <a rel="nofollow" class="external text" href="http://rcgldr.net/misc/ecc.pdf">CRC and Reed Solomon ECC</a> (PDF)</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=CRC+and+Reed+Solomon+ECC&rft.date=1995-04&rft.aulast=Reid&rft.aufirst=Jeff+A.&rft_id=http%3A%2F%2Frcgldr.net%2Fmisc%2Fecc.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReed%E2%80%93Solomon+error+correction" class="Z3988"></li></ul> <div class="mw-heading mw-heading3"><h3 id="Implementations">Implementations</h3>[<a href="/w/index.php?title=Reed%E2%80%93Solomon_error_correction&action=edit&section=48" title="Edit section: Implementations">edit</a>]</div> <ul><li><a rel="nofollow" class="external text" href="https://www.ka9q.net/code/fec/">FEC library in C by Phil Karn (aka KA9Q) includes Reed–Solomon codec, both arbitrary and optimized (223,255) version</a></li> <li><a rel="nofollow" class="external text" href="https://code.google.com/p/rssoft/">Open Source C++ Reed–Solomon Soft Decoding library</a></li> <li><a rel="nofollow" class="external text" href="http://dept.ee.wits.ac.za/~versfeld/research_resources/sourcecode/Errors_And_Erasures_Test.zip">Matlab implementation of errors and-erasures Reed–Solomon decoding</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; 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Reed–Solomon error correction - Wikipedia