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<span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Principles" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Principles"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Principles</span> </div> </a> <ul id="toc-Principles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Types_of_error_correction" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Types_of_error_correction"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Types of error correction</span> </div> </a> <button aria-controls="toc-Types_of_error_correction-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Types of error correction subsection</span> </button> <ul id="toc-Types_of_error_correction-sublist" class="vector-toc-list"> <li id="toc-Automatic_repeat_request" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Automatic_repeat_request"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Automatic repeat request</span> </div> </a> <ul id="toc-Automatic_repeat_request-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Forward_error_correction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Forward_error_correction"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Forward error correction</span> </div> </a> <ul id="toc-Forward_error_correction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hybrid_schemes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hybrid_schemes"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Hybrid schemes</span> </div> </a> <ul id="toc-Hybrid_schemes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Types_of_error_detection" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Types_of_error_detection"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Types of error detection</span> </div> </a> <button aria-controls="toc-Types_of_error_detection-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Types of error detection subsection</span> </button> <ul id="toc-Types_of_error_detection-sublist" class="vector-toc-list"> <li id="toc-Minimum_distance_coding" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Minimum_distance_coding"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Minimum distance coding</span> </div> </a> <ul id="toc-Minimum_distance_coding-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Repetition_codes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Repetition_codes"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Repetition codes</span> </div> </a> <ul id="toc-Repetition_codes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Parity_bit" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Parity_bit"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Parity bit</span> </div> </a> <ul id="toc-Parity_bit-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Checksum" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Checksum"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Checksum</span> </div> </a> <ul id="toc-Checksum-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cyclic_redundancy_check" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cyclic_redundancy_check"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>Cyclic redundancy check</span> </div> </a> <ul id="toc-Cyclic_redundancy_check-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cryptographic_hash_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cryptographic_hash_function"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.6</span> <span>Cryptographic hash function</span> </div> </a> <ul id="toc-Cryptographic_hash_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Digital_signature" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Digital_signature"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.7</span> <span>Digital signature</span> </div> </a> <ul id="toc-Digital_signature-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Error_correction_code" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Error_correction_code"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.8</span> <span>Error correction code</span> </div> </a> <ul id="toc-Error_correction_code-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Applications</span> </div> </a> <button aria-controls="toc-Applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Applications subsection</span> </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Internet" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Internet"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Internet</span> </div> </a> <ul id="toc-Internet-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Deep-space_telecommunications" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Deep-space_telecommunications"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Deep-space telecommunications</span> </div> </a> <ul id="toc-Deep-space_telecommunications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Satellite_broadcasting" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Satellite_broadcasting"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Satellite broadcasting</span> </div> </a> <ul id="toc-Satellite_broadcasting-sublist" class="vector-toc-list"> </ul> </li> <li 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"> <span class="vector-toc-numb">6.4</span> <span>Data storage</span> </div> </a> <ul id="toc-Data_storage-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Error-correcting_memory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Error-correcting_memory"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5</span> <span>Error-correcting memory</span> </div> </a> <ul id="toc-Error-correcting_memory-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </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" > <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" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </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"><span class="mw-page-title-main">Error detection and correction</span></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 35 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-35" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">35 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%83%D8%B4%D9%81_%D8%A7%D9%84%D8%A3%D8%AE%D8%B7%D8%A7%D8%A1_%D9%88%D8%AA%D8%B5%D8%AD%D9%8A%D8%AD%D9%87%D8%A7" title="كشف الأخطاء وتصحيحها – Arabic" lang="ar" hreflang="ar" data-title="كشف الأخطاء وتصحيحها" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Deteici%C3%B3n_y_correici%C3%B3n_d%27errores" title="Deteición y correición d'errores – Asturian" lang="ast" hreflang="ast" data-title="Deteición y correición d'errores" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/X%C9%99ta_a%C5%9Fkarlama_kodla%C5%9Fd%C4%B1rmas%C4%B1" title="Xəta aşkarlama kodlaşdırması – Azerbaijani" lang="az" hreflang="az" data-title="Xəta aşkarlama kodlaşdırması" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%92%D1%8B%D1%8F%D1%9E%D0%BB%D0%B5%D0%BD%D0%BD%D0%B5_%D1%96_%D0%B2%D1%8B%D0%BF%D1%80%D0%B0%D1%9E%D0%BB%D0%B5%D0%BD%D0%BD%D0%B5_%D0%BF%D0%B0%D0%BC%D1%8B%D0%BB%D0%B0%D0%BA" title="Выяўленне і выпраўленне памылак – Belarusian" lang="be" hreflang="be" data-title="Выяўленне і выпраўленне памылак" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9A%D0%B0%D1%80%D1%8D%D0%BA%D1%86%D1%8B%D1%8F_%D0%BF%D0%B0%D0%BC%D1%8B%D0%BB%D0%B0%D0%BA" title="Карэкцыя памылак – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Карэкцыя памылак" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Detector_i_corrector_d%27errors" title="Detector i corrector d'errors – Catalan" lang="ca" hreflang="ca" data-title="Detector i corrector d'errors" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Detekce_a_oprava_chyb" title="Detekce a oprava chyb – Czech" lang="cs" hreflang="cs" data-title="Detekce a oprava chyb" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Fehlerkorrekturverfahren" title="Fehlerkorrekturverfahren – German" lang="de" hreflang="de" data-title="Fehlerkorrekturverfahren" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CE%BD%CE%AF%CF%87%CE%BD%CE%B5%CF%85%CF%83%CE%B7_%CE%BA%CE%B1%CE%B9_%CE%B4%CE%B9%CF%8C%CF%81%CE%B8%CF%89%CF%83%CE%B7_%CF%83%CF%86%CE%B1%CE%BB%CE%BC%CE%AC%CF%84%CF%89%CE%BD" title="Ανίχνευση και διόρθωση σφαλμάτων – Greek" lang="el" hreflang="el" data-title="Ανίχνευση και διόρθωση σφαλμάτων" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Detecci%C3%B3n_y_correcci%C3%B3n_de_errores" title="Detección y corrección de errores – Spanish" lang="es" hreflang="es" data-title="Detección y corrección de errores" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Errore-detekzio" title="Errore-detekzio – Basque" lang="eu" hreflang="eu" data-title="Errore-detekzio" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%B4%D8%AE%DB%8C%D8%B5_%D9%88_%D8%AA%D8%B5%D8%AD%DB%8C%D8%AD_%D8%AE%D8%B7%D8%A7" title="تشخیص و تصحیح خطا – Persian" lang="fa" hreflang="fa" data-title="تشخیص و تصحیح خطا" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%98%A4%EB%A5%98_%EA%B2%80%EC%B6%9C_%EC%A0%95%EC%A0%95" title="오류 검출 정정 – Korean" lang="ko" hreflang="ko" data-title="오류 검출 정정" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A5%81%E0%A4%9F%E0%A4%BF_%E0%A4%B8%E0%A4%82%E0%A4%B8%E0%A5%82%E0%A4%9A%E0%A4%A8_%E0%A4%94%E0%A4%B0_%E0%A4%B8%E0%A5%81%E0%A4%A7%E0%A4%BE%E0%A4%B0" title="त्रुटि संसूचन और सुधार – Hindi" lang="hi" hreflang="hi" data-title="त्रुटि संसूचन और सुधार" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Rilevazione_e_correzione_d%27errore" title="Rilevazione e correzione d'errore – Italian" lang="it" hreflang="it" data-title="Rilevazione e correzione d'errore" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Ugunduzi_na_usahihishaji_wa_makosa" title="Ugunduzi na usahihishaji wa makosa – Swahili" lang="sw" hreflang="sw" data-title="Ugunduzi na usahihishaji wa makosa" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Deprehensio_et_correctio_mendorum" title="Deprehensio et correctio mendorum – Latin" lang="la" hreflang="la" data-title="Deprehensio et correctio mendorum" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/K%C4%BC%C5%ABdu_kontrole" title="Kļūdu kontrole – Latvian" lang="lv" hreflang="lv" data-title="Kļūdu kontrole" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Hibajav%C3%ADt%C3%A1s" title="Hibajavítás – Hungarian" lang="hu" hreflang="hu" data-title="Hibajavítás" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Foutdetectie_en_-correctie" title="Foutdetectie en -correctie – Dutch" lang="nl" hreflang="nl" data-title="Foutdetectie en -correctie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%AA%A4%E3%82%8A%E6%A4%9C%E5%87%BA%E8%A8%82%E6%AD%A3" title="誤り検出訂正 – Japanese" lang="ja" hreflang="ja" data-title="誤り検出訂正" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Feildeteksjon_og_korreksjon" title="Feildeteksjon og korreksjon – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Feildeteksjon og korreksjon" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Kodowanie_korekcyjne" title="Kodowanie korekcyjne – Polish" lang="pl" hreflang="pl" data-title="Kodowanie korekcyjne" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Detec%C3%A7%C3%A3o_e_corre%C3%A7%C3%A3o_de_erros" title="Detecção e correção de erros – Portuguese" lang="pt" hreflang="pt" data-title="Detecção e correção de erros" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BD%D1%82%D1%80%D0%BE%D0%BB%D1%8C_%D0%BE%D1%88%D0%B8%D0%B1%D0%BE%D0%BA" title="Контроль ошибок – Russian" lang="ru" hreflang="ru" data-title="Контроль ошибок" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Error_detection_and_correction" 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class="vector-body" aria-labelledby="firstHeading" 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"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=EDAC_(Linux)&redirect=no" class="mw-redirect" title="EDAC (Linux)">EDAC (Linux)</a>)</span></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">Techniques that enable reliable delivery of digital data over unreliable communication channels</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 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For knowledge, see <a href="/wiki/Fact-checking" title="Fact-checking">Fact-checking</a> and <a href="/wiki/Problem_solving" title="Problem solving">Problem solving</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-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><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></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Error_detection_and_correction" title="Special:EditPage/Error detection and correction">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a> in this article. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Error+detection+and+correction%22">"Error detection and correction"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Error+detection+and+correction%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Error+detection+and+correction%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Error+detection+and+correction%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Error+detection+and+correction%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Error+detection+and+correction%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">August 2008</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Reed%E2%80%93Solomon_error_correction_Mona_Lisa_LroLrLasercomFig4.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Reed%E2%80%93Solomon_error_correction_Mona_Lisa_LroLrLasercomFig4.jpg/220px-Reed%E2%80%93Solomon_error_correction_Mona_Lisa_LroLrLasercomFig4.jpg" decoding="async" width="220" height="133" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Reed%E2%80%93Solomon_error_correction_Mona_Lisa_LroLrLasercomFig4.jpg/330px-Reed%E2%80%93Solomon_error_correction_Mona_Lisa_LroLrLasercomFig4.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Reed%E2%80%93Solomon_error_correction_Mona_Lisa_LroLrLasercomFig4.jpg/440px-Reed%E2%80%93Solomon_error_correction_Mona_Lisa_LroLrLasercomFig4.jpg 2x" data-file-width="956" data-file-height="576" /></a><figcaption>To clean up transmission errors introduced by Earth's atmosphere (left), Goddard scientists applied Reed–Solomon error correction (right), which is commonly used in CDs and DVDs. Typical errors include missing pixels (white) and false signals (black). The white stripe indicates a brief period when transmission was interrupted.</figcaption></figure> <p>In <a href="/wiki/Information_theory" title="Information theory">information theory</a> and <a href="/wiki/Coding_theory" title="Coding theory">coding theory</a> with applications in <a href="/wiki/Computer_science" title="Computer science">computer science</a> and <a href="/wiki/Telecommunications" title="Telecommunications">telecommunications</a>, <b>error detection and correction</b> (<b>EDAC</b>) or <b>error control</b> are techniques that enable <a href="/wiki/Reliable_delivery" class="mw-redirect" title="Reliable delivery">reliable delivery</a> of <a href="/wiki/Digital_data" title="Digital data">digital data</a> over unreliable <a href="/wiki/Communication_channel" title="Communication channel">communication channels</a>. Many communication channels are subject to <a href="/wiki/Channel_noise" class="mw-redirect" title="Channel noise">channel noise</a>, and thus errors may be introduced during transmission from the source to a receiver. Error detection techniques allow detecting such errors, while error correction enables reconstruction of the original data in many cases. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Error_detection_and_correction&action=edit&section=1" title="Edit section: Definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>Error detection</i> is the detection of errors caused by noise or other impairments during transmission from the transmitter to the receiver. </p><p><i>Error correction</i> is the detection of errors and reconstruction of the original, error-free data. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Error_detection_and_correction&action=edit&section=2" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In classical antiquity, <a href="/wiki/Copyist" title="Copyist">copyists</a> of the <a href="/wiki/Hebrew_Bible" title="Hebrew Bible">Hebrew Bible</a> were paid for their work according to the number of <a href="https://en.wiktionary.org/wiki/stich" class="extiw" title="wiktionary:stich">stichs</a> (lines of verse). As the prose books of the Bible were hardly ever written in stichs, the copyists, in order to estimate the amount of work, had to count the letters.<sup id="cite_ref-Jewish_1-0" class="reference"><a href="#cite_note-Jewish-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> This also helped ensure accuracy in the transmission of the text with the production of subsequent copies.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Between the 7th and 10th centuries CE a <a href="/wiki/Masoretes" title="Masoretes">group of Jewish scribes</a> formalized and expanded this to create the <a href="/wiki/Masoretic_Text#Numerical_Masorah" title="Masoretic Text">Numerical Masorah</a> to ensure accurate reproduction of the sacred text. It included counts of the number of words in a line, section, book and groups of books, noting the middle stich of a book, word use statistics, and commentary.<sup id="cite_ref-Jewish_1-1" class="reference"><a href="#cite_note-Jewish-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Standards became such that a deviation in even a single letter in a Torah scroll was considered unacceptable.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> The effectiveness of their error correction method was verified by the accuracy of copying through the centuries demonstrated by discovery of the <a href="/wiki/Dead_Sea_Scrolls" title="Dead Sea Scrolls">Dead Sea Scrolls</a> in 1947–1956, dating from <abbr title="circa">c.</abbr><span style="white-space:nowrap;"> 150 BCE-75 CE</span>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>The modern development of <a href="/wiki/Error_correction_code" title="Error correction code">error correction codes</a> is credited to <a href="/wiki/Richard_Hamming" title="Richard Hamming">Richard Hamming</a> in 1947.<sup id="cite_ref-Thompson_6-0" class="reference"><a href="#cite_note-Thompson-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> A description of <a href="/wiki/Hamming%27s_code" class="mw-redirect" title="Hamming's code">Hamming's code</a> appeared in <a href="/wiki/Claude_Shannon" title="Claude Shannon">Claude Shannon</a>'s <i>A Mathematical Theory of Communication</i><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> and was quickly generalized by <a href="/wiki/Marcel_J._E._Golay" title="Marcel J. E. Golay">Marcel J. E. Golay</a>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Principles">Principles</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Error_detection_and_correction&action=edit&section=3" title="Edit section: Principles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>All error-detection and correction schemes add some <a href="/wiki/Redundancy_(information_theory)" title="Redundancy (information theory)">redundancy</a> (i.e., some extra data) to a message, which receivers can use to check consistency of the delivered message and to recover data that has been determined to be corrupted. Error detection and correction schemes can be either <a href="/wiki/Systematic_code" title="Systematic code">systematic</a> or non-systematic. In a systematic scheme, the transmitter sends the original (error-free) data and attaches a fixed number of <i>check bits</i> (or <i>parity data</i>), which are derived from the data bits by some encoding algorithm. If error detection is required, a receiver can simply apply the same algorithm to the received data bits and compare its output with the received check bits; if the values do not match, an error has occurred at some point during the transmission. If error correction is required, a receiver can apply the decoding algorithm to the received data bits and the received check bits to recover the original error-free data. In a system that uses a non-systematic code, the original message is transformed into an encoded message carrying the same information and that has at least as many bits as the original message. </p><p>Good error control performance requires the scheme to be selected based on the characteristics of the communication channel. Common <a href="/wiki/Channel_model" class="mw-redirect" title="Channel model">channel models</a> include <a href="/wiki/Memoryless" class="mw-redirect" title="Memoryless">memoryless</a> models where errors occur randomly and with a certain probability, and dynamic models where errors occur primarily in <a href="/wiki/Burst_error" title="Burst error">bursts</a>. Consequently, error-detecting and -correcting codes can be generally distinguished between <i>random-error-detecting/correcting</i> and <i>burst-error-detecting/correcting</i>. Some codes can also be suitable for a mixture of random errors and burst errors. </p><p>If the channel characteristics cannot be determined, or are highly variable, an error-detection scheme may be combined with a system for retransmissions of erroneous data. This is known as <a href="/wiki/Automatic_repeat_request" title="Automatic repeat request">automatic repeat request</a> (ARQ), and is most notably used in the Internet. An alternate approach for error control is <a href="/wiki/Hybrid_automatic_repeat_request" title="Hybrid automatic repeat request">hybrid automatic repeat request</a> (HARQ), which is a combination of ARQ and error-correction coding. </p> <div class="mw-heading mw-heading2"><h2 id="Types_of_error_correction">Types of error correction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Error_detection_and_correction&action=edit&section=4" title="Edit section: Types of error correction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are three major types of error correction:<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Automatic_repeat_request">Automatic repeat request</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Error_detection_and_correction&action=edit&section=5" title="Edit section: Automatic repeat request"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Automatic_repeat_request" title="Automatic repeat request">Automatic repeat request</a> (ARQ) is an error control method for data transmission that makes use of error-detection codes, acknowledgment and/or negative acknowledgment messages, and <a href="/wiki/Timeout_(computing)" title="Timeout (computing)">timeouts</a> to achieve reliable data transmission. An <i>acknowledgment</i> is a message sent by the receiver to indicate that it has correctly received a <a href="/wiki/Data_frame" class="mw-redirect" title="Data frame">data frame</a>. </p><p>Usually, when the transmitter does not receive the acknowledgment before the timeout occurs (i.e., within a reasonable amount of time after sending the data frame), it retransmits the frame until it is either correctly received or the error persists beyond a predetermined number of retransmissions. </p><p>Three types of ARQ protocols are <a href="/wiki/Stop-and-wait_ARQ" title="Stop-and-wait ARQ">Stop-and-wait ARQ</a>, <a href="/wiki/Go-Back-N_ARQ" title="Go-Back-N ARQ">Go-Back-N ARQ</a>, and <a href="/wiki/Selective_Repeat_ARQ" title="Selective Repeat ARQ">Selective Repeat ARQ</a>. </p><p>ARQ is appropriate if the communication channel has varying or unknown <a href="/wiki/Channel_capacity" title="Channel capacity">capacity</a>, such as is the case on the Internet. However, ARQ requires the availability of a <a href="/wiki/Backward_channel" title="Backward channel">back channel</a>, results in possibly increased <a href="/wiki/Latency_(engineering)" title="Latency (engineering)">latency</a> due to retransmissions, and requires the maintenance of buffers and timers for retransmissions, which in the case of <a href="/wiki/Network_congestion" title="Network congestion">network congestion</a> can put a strain on the server and overall network capacity.<sup id="cite_ref-reliable-erasure-code_10-0" class="reference"><a href="#cite_note-reliable-erasure-code-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>For example, ARQ is used on shortwave radio data links in the form of <a href="/wiki/ARQ-E" title="ARQ-E">ARQ-E</a>, or combined with multiplexing as <a href="/wiki/ARQ-M" title="ARQ-M">ARQ-M</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Forward_error_correction">Forward error correction</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Error_detection_and_correction&action=edit&section=6" title="Edit section: Forward error correction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Forward_error_correction" class="mw-redirect" title="Forward error correction">Forward error correction</a> (FEC) is a process of adding <a href="/wiki/Redundancy_(information_theory)" title="Redundancy (information theory)">redundant data</a> such as an <a href="/wiki/Error-correcting_code" class="mw-redirect" title="Error-correcting code">error-correcting code</a> (ECC) to a message so that it can be recovered by a receiver even when a number of errors (up to the capability of the code being used) are introduced, either during the process of transmission or on storage. Since the receiver does not have to ask the sender for retransmission of the data, a <a href="/wiki/Backchannel" title="Backchannel">backchannel</a> is not required in forward error correction. Error-correcting codes are used in <a href="/wiki/Physical_layer" title="Physical layer">lower-layer</a> communication such as <a href="/wiki/Cellular_network" title="Cellular network">cellular network</a>, high-speed <a href="/wiki/Fiber-optic_communication" title="Fiber-optic communication">fiber-optic communication</a> and <a href="/wiki/Wi-Fi" title="Wi-Fi">Wi-Fi</a>,<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> as well as for reliable storage in media such as <a href="/wiki/Flash_memory" title="Flash memory">flash memory</a>, <a href="/wiki/Hard_disk" class="mw-redirect" title="Hard disk">hard disk</a> and <a href="/wiki/ECC_memory" title="ECC memory">RAM</a>.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>Error-correcting codes are usually distinguished between <a href="/wiki/Convolutional_code" title="Convolutional code">convolutional codes</a> and <a href="/wiki/Block_code" title="Block code">block codes</a>: </p> <ul><li><i>Convolutional codes</i> are processed on a bit-by-bit basis. They are particularly suitable for implementation in hardware, and the <a href="/wiki/Viterbi_decoder" title="Viterbi decoder">Viterbi decoder</a> allows <a href="/wiki/Maximum_likelihood_decoding" class="mw-redirect" title="Maximum likelihood decoding">optimal decoding</a>.</li> <li><i>Block codes</i> are processed on a <a href="/wiki/Block_(telecommunications)" title="Block (telecommunications)">block-by-block</a> basis. Early examples of block codes are <a href="/wiki/Repetition_code" title="Repetition code">repetition codes</a>, <a href="/wiki/Hamming_code" title="Hamming code">Hamming codes</a> and <a href="/wiki/Multidimensional_parity-check_code" title="Multidimensional parity-check code">multidimensional parity-check codes</a>. They were followed by a number of efficient codes, <a href="/wiki/Reed%E2%80%93Solomon_code" class="mw-redirect" title="Reed–Solomon code">Reed–Solomon codes</a> being the most notable due to their current widespread use. <a href="/wiki/Turbo_code" title="Turbo code">Turbo codes</a> and <a href="/wiki/Low-density_parity-check_code" title="Low-density parity-check code">low-density parity-check codes</a> (LDPC) are relatively new constructions that can provide almost <a href="/wiki/Category:Capacity-approaching_codes" title="Category:Capacity-approaching codes">optimal efficiency</a>.</li></ul> <p><a href="/wiki/Shannon%27s_theorem" class="mw-redirect" title="Shannon's theorem">Shannon's theorem</a> is an important theorem in forward error correction, and describes the maximum <a href="/wiki/Information_rate" class="mw-redirect" title="Information rate">information rate</a> at which reliable communication is possible over a channel that has a certain error probability or <a href="/wiki/Signal-to-noise_ratio" title="Signal-to-noise ratio">signal-to-noise ratio</a> (SNR). This strict upper limit is expressed in terms of the <a href="/wiki/Channel_capacity" title="Channel capacity">channel capacity</a>. More specifically, the theorem says that there exist codes such that with increasing encoding length the probability of error on a <a href="/wiki/Channel_model" class="mw-redirect" title="Channel model">discrete memoryless channel</a> can be made arbitrarily small, provided that the <a href="/wiki/Code_rate" title="Code rate">code rate</a> is smaller than the channel capacity. The code rate is defined as the fraction <i>k/n</i> of <i>k</i> source symbols and <i>n</i> encoded symbols. </p><p>The actual maximum code rate allowed depends on the error-correcting code used, and may be lower. This is because Shannon's proof was only of existential nature, and did not show how to construct codes that are both optimal and have <a href="/wiki/Polynomial_time" class="mw-redirect" title="Polynomial time">efficient</a> encoding and decoding algorithms. </p> <div class="mw-heading mw-heading3"><h3 id="Hybrid_schemes">Hybrid schemes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Error_detection_and_correction&action=edit&section=7" title="Edit section: Hybrid schemes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Hybrid_ARQ" class="mw-redirect" title="Hybrid ARQ">Hybrid ARQ</a></div> <p><a href="/wiki/Hybrid_ARQ" class="mw-redirect" title="Hybrid ARQ">Hybrid ARQ</a> is a combination of ARQ and forward error correction. There are two basic approaches:<sup id="cite_ref-reliable-erasure-code_10-1" class="reference"><a href="#cite_note-reliable-erasure-code-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <ul><li>Messages are always transmitted with FEC parity data (and error-detection redundancy). A receiver decodes a message using the parity information and requests retransmission using ARQ only if the parity data was not sufficient for successful decoding (identified through a failed integrity check).</li> <li>Messages are transmitted without parity data (only with error-detection information). If a receiver detects an error, it requests FEC information from the transmitter using ARQ and uses it to reconstruct the original message.</li></ul> <p>The latter approach is particularly attractive on an <a href="/wiki/Erasure_channel" class="mw-redirect" title="Erasure channel">erasure channel</a> when using a <a href="/wiki/Rateless_erasure_code" class="mw-redirect" title="Rateless erasure code">rateless erasure code</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Types_of_error_detection">Types of error detection</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Error_detection_and_correction&action=edit&section=8" title="Edit section: Types of error detection"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Error detection is most commonly realized using a suitable <a href="/wiki/Hash_function" title="Hash function">hash function</a> (or specifically, a <a href="/wiki/Checksum" title="Checksum">checksum</a>, <a href="/wiki/Cyclic_redundancy_check" title="Cyclic redundancy check">cyclic redundancy check</a> or other algorithm). A hash function adds a fixed-length <i>tag</i> to a message, which enables receivers to verify the delivered message by recomputing the tag and comparing it with the one provided. </p><p>There exists a vast variety of different hash function designs. However, some are of particularly widespread use because of either their simplicity or their suitability for detecting certain kinds of errors (e.g., the cyclic redundancy check's performance in detecting <a href="/wiki/Burst_error" title="Burst error">burst errors</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Minimum_distance_coding">Minimum distance coding</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Error_detection_and_correction&action=edit&section=9" title="Edit section: Minimum distance coding"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A random-error-correcting code based on <a href="/wiki/Minimum_distance_coding" class="mw-redirect" title="Minimum distance coding">minimum distance coding</a> can provide a strict guarantee on the number of detectable errors, but it may not protect against a <a href="/wiki/Preimage_attack" title="Preimage attack">preimage attack</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Repetition_codes">Repetition codes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Error_detection_and_correction&action=edit&section=10" title="Edit section: Repetition codes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Repetition_code" title="Repetition code">repetition code</a> is a coding scheme that repeats the bits across a channel to achieve error-free communication. Given a stream of data to be transmitted, the data are divided into blocks of bits. Each block is transmitted some predetermined number of times. For example, to send the bit pattern <style data-mw-deduplicate="TemplateStyles:r886049734">.mw-parser-output .monospaced{font-family:monospace,monospace}</style><span class="monospaced">1011</span>, the four-bit block can be repeated three times, thus producing <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">1011 1011 1011</span>. If this twelve-bit pattern was received as <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">1010 1011 1011</span> – where the first block is unlike the other two – an error has occurred. </p><p>A repetition code is very inefficient and can be susceptible to problems if the error occurs in exactly the same place for each group (e.g., <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">1010 1010 1010</span> in the previous example would be detected as correct). The advantage of repetition codes is that they are extremely simple, and are in fact used in some transmissions of <a href="/wiki/Numbers_station" title="Numbers station">numbers stations</a>.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Parity_bit">Parity bit</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Error_detection_and_correction&action=edit&section=11" title="Edit section: Parity bit"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Parity_bit" title="Parity bit">Parity bit</a></div> <p>A <i>parity bit</i> is a bit that is added to a group of source bits to ensure that the number of set bits (i.e., bits with value 1) in the outcome is even or odd. It is a very simple scheme that can be used to detect single or any other odd number (i.e., three, five, etc.) of errors in the output. An even number of flipped bits will make the parity bit appear correct even though the data is erroneous. </p><p>Parity bits added to each <i>word</i> sent are called <a href="/wiki/Transverse_redundancy_check" title="Transverse redundancy check">transverse redundancy checks</a>, while those added at the end of a stream of <i>words</i> are called <a href="/wiki/Longitudinal_redundancy_check" title="Longitudinal redundancy check">longitudinal redundancy checks</a>. For example, if each of a series of m-bit <i>words</i> has a parity bit added, showing whether there were an odd or even number of ones in that word, any word with a single error in it will be detected. It will not be known where in the word the error is, however. If, in addition, after each stream of n words a parity sum is sent, each bit of which shows whether there were an odd or even number of ones at that bit-position sent in the most recent group, the exact position of the error can be determined and the error corrected. This method is only guaranteed to be effective, however, if there are no more than 1 error in every group of n words. With more error correction bits, more errors can be detected and in some cases corrected. </p><p>There are also other bit-grouping techniques. </p> <div class="mw-heading mw-heading3"><h3 id="Checksum">Checksum</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Error_detection_and_correction&action=edit&section=12" title="Edit section: Checksum"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Checksum" title="Checksum">Checksum</a></div> <p>A <i>checksum</i> of a message is a <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modular arithmetic</a> sum of message code words of a fixed word length (e.g., byte values). The sum may be negated by means of a <a href="/wiki/Ones%27-complement" class="mw-redirect" title="Ones'-complement">ones'-complement</a> operation prior to transmission to detect unintentional all-zero messages. </p><p>Checksum schemes include parity bits, <a href="/wiki/Check_digit" title="Check digit">check digits</a>, and <a href="/wiki/Longitudinal_redundancy_check" title="Longitudinal redundancy check">longitudinal redundancy checks</a>. Some checksum schemes, such as the <a href="/wiki/Damm_algorithm" title="Damm algorithm">Damm algorithm</a>, the <a href="/wiki/Luhn_algorithm" title="Luhn algorithm">Luhn algorithm</a>, and the <a href="/wiki/Verhoeff_algorithm" title="Verhoeff algorithm">Verhoeff algorithm</a>, are specifically designed to detect errors commonly introduced by humans in writing down or remembering identification numbers. </p> <div class="mw-heading mw-heading3"><h3 id="Cyclic_redundancy_check">Cyclic redundancy check</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Error_detection_and_correction&action=edit&section=13" title="Edit section: Cyclic redundancy check"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Cyclic_redundancy_check" title="Cyclic redundancy check">Cyclic redundancy check</a></div> <p>A <i>cyclic redundancy check</i> (CRC) is a non-secure <a href="/wiki/Hash_function" title="Hash function">hash function</a> designed to detect accidental changes to digital data in computer networks. It is not suitable for detecting maliciously introduced errors. It is characterized by specification of a <i>generator polynomial</i>, which is used as the <a href="/wiki/Divisor" title="Divisor">divisor</a> in a <a href="/wiki/Polynomial_long_division" title="Polynomial long division">polynomial long division</a> over a <a href="/wiki/Finite_field" title="Finite field">finite field</a>, taking the input data as the <a href="/wiki/Dividend" title="Dividend">dividend</a>. The <a href="/wiki/Remainder" title="Remainder">remainder</a> becomes the result. </p><p>A CRC has properties that make it well suited for detecting <a href="/wiki/Burst_error" title="Burst error">burst errors</a>. CRCs are particularly easy to implement in hardware and are therefore commonly used in <a href="/wiki/Computer_network" title="Computer network">computer networks</a> and storage devices such as <a href="/wiki/Hard_disk_drives" class="mw-redirect" title="Hard disk drives">hard disk drives</a>. </p><p>The parity bit can be seen as a special-case 1-bit CRC. </p> <div class="mw-heading mw-heading3"><h3 id="Cryptographic_hash_function">Cryptographic hash function</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Error_detection_and_correction&action=edit&section=14" title="Edit section: Cryptographic hash function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Cryptographic_hash_function" title="Cryptographic hash function">Cryptographic hash function</a></div> <p>The output of a <i>cryptographic hash function</i>, also known as a <i>message digest</i>, can provide strong assurances about <a href="/wiki/Data_integrity" title="Data integrity">data integrity</a>, whether changes of the data are accidental (e.g., due to transmission errors) or maliciously introduced. Any modification to the data will likely be detected through a mismatching hash value. Furthermore, given some hash value, it is typically infeasible to find some input data (other than the one given) that will yield the same hash value. If an attacker can change not only the message but also the hash value, then a <i>keyed hash</i> or <a href="/wiki/Message_authentication_code" title="Message authentication code">message authentication code</a> (MAC) can be used for additional security. Without knowing the key, it is not possible for the attacker to easily or conveniently calculate the correct keyed hash value for a modified message. </p> <div class="mw-heading mw-heading3"><h3 id="Digital_signature">Digital signature</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Error_detection_and_correction&action=edit&section=15" title="Edit section: Digital signature"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Digital_signature" title="Digital signature">Digital signature</a></div> <p>Digital signatures can provide strong assurances about data integrity, whether the changes of the data are accidental or maliciously introduced. Digital signatures are perhaps most notable for being part of the HTTPS protocol for securely browsing the web. </p> <div class="mw-heading mw-heading3"><h3 id="Error_correction_code">Error correction code</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Error_detection_and_correction&action=edit&section=16" title="Edit section: Error correction code"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Error_correction_code" title="Error correction code">Error correction code</a></div> <p>Any error-correcting code can be used for error detection. A code with <i>minimum <a href="/wiki/Hamming_distance" title="Hamming distance">Hamming distance</a></i>, <i>d</i>, can detect up to <i>d</i> − 1 errors in a code word. Using minimum-distance-based error-correcting codes for error detection can be suitable if a strict limit on the minimum number of errors to be detected is desired. </p><p>Codes with minimum Hamming distance <i>d</i> = 2 are degenerate cases of error-correcting codes and can be used to detect single errors. The parity bit is an example of a single-error-detecting code. </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Error_detection_and_correction&action=edit&section=17" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Applications that require low latency (such as telephone conversations) cannot use <a href="/wiki/Automatic_repeat_request" title="Automatic repeat request">automatic repeat request</a> (ARQ); they must use <a href="/wiki/Forward_error_correction" class="mw-redirect" title="Forward error correction">forward error correction</a> (FEC). By the time an ARQ system discovers an error and re-transmits it, the re-sent data will arrive too late to be usable. </p><p>Applications where the transmitter immediately forgets the information as soon as it is sent (such as most television cameras) cannot use ARQ; they must use FEC because when an error occurs, the original data is no longer available. </p><p>Applications that use ARQ must have a <a href="/wiki/Return_channel" title="Return channel">return channel</a>; applications having no return channel cannot use ARQ. </p><p>Applications that require extremely low error rates (such as digital money transfers) must use ARQ due to the possibility of uncorrectable errors with FEC. </p><p>Reliability and inspection engineering also make use of the theory of error-correcting codes.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Internet">Internet</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Error_detection_and_correction&action=edit&section=18" title="Edit section: Internet"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a typical <a href="/wiki/TCP/IP" class="mw-redirect" title="TCP/IP">TCP/IP</a> stack, error control is performed at multiple levels: </p> <ul><li>Each <a href="/wiki/Ethernet_frame" title="Ethernet frame">Ethernet frame</a> uses <a href="/wiki/Cyclic_redundancy_check" title="Cyclic redundancy check">CRC-32</a> error detection. Frames with detected errors are discarded by the receiver hardware.</li> <li>The <a href="/wiki/IPv4" title="IPv4">IPv4</a> header contains a <a href="/wiki/IPv4_header_checksum" class="mw-redirect" title="IPv4 header checksum">checksum</a> protecting the contents of the header. <a href="/wiki/Network_packet" title="Network packet">Packets</a> with incorrect checksums are dropped within the network or at the receiver.</li> <li>The checksum was omitted from the <a href="/wiki/IPv6" title="IPv6">IPv6</a> header in order to minimize processing costs in <a href="/wiki/Network_routing" class="mw-redirect" title="Network routing">network routing</a> and because current <a href="/wiki/Link_layer" title="Link layer">link layer</a> technology is assumed to provide sufficient error detection (see also RFC 3819).</li> <li><a href="/wiki/User_Datagram_Protocol" title="User Datagram Protocol">UDP</a> has an optional checksum covering the payload and addressing information in the UDP and IP headers. Packets with incorrect checksums are discarded by the <a href="/wiki/Network_stack" class="mw-redirect" title="Network stack">network stack</a>. The checksum is optional under IPv4, and required under IPv6. When omitted, it is assumed the data-link layer provides the desired level of error protection.</li> <li><a href="/wiki/Transmission_Control_Protocol" title="Transmission Control Protocol">TCP</a> provides a checksum for protecting the payload and addressing information in the TCP and IP headers. Packets with incorrect checksums are discarded by the network stack and eventually get retransmitted using ARQ, either explicitly (such as through <a href="/wiki/Three-way_handshake" class="mw-redirect" title="Three-way handshake">three-way handshake</a>) or implicitly due to a <a href="/wiki/Timeout_(computing)" title="Timeout (computing)">timeout</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Deep-space_telecommunications">Deep-space telecommunications</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Error_detection_and_correction&action=edit&section=19" title="Edit section: Deep-space telecommunications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The development of error-correction codes was tightly coupled with the history of deep-space missions due to the extreme dilution of signal power over interplanetary distances, and the limited power availability aboard space probes. Whereas early missions sent their data uncoded, starting in 1968, digital error correction was implemented in the form of (sub-optimally decoded) <a href="/wiki/Convolutional_code" title="Convolutional code">convolutional codes</a> and <a href="/wiki/Reed%E2%80%93Muller_code" title="Reed–Muller code">Reed–Muller codes</a>.<sup id="cite_ref-deep-space-codes_17-0" class="reference"><a href="#cite_note-deep-space-codes-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> The Reed–Muller code was well suited to the noise the spacecraft was subject to (approximately matching a <a href="/wiki/Gaussian_function" title="Gaussian function">bell curve</a>), and was implemented for the Mariner spacecraft and used on missions between 1969 and 1977. </p><p>The <a href="/wiki/Voyager_1" title="Voyager 1">Voyager 1</a> and <a href="/wiki/Voyager_2" title="Voyager 2">Voyager 2</a> missions, which started in 1977, were designed to deliver color imaging and scientific information from <a href="/wiki/Jupiter" title="Jupiter">Jupiter</a> and <a href="/wiki/Saturn" title="Saturn">Saturn</a>.<sup id="cite_ref-voyager_18-0" class="reference"><a href="#cite_note-voyager-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> This resulted in increased coding requirements, and thus, the spacecraft were supported by (optimally <a href="/wiki/Viterbi_decoder" title="Viterbi decoder">Viterbi-decoded</a>) convolutional codes that could be <a href="/wiki/Concatenated_code" class="mw-redirect" title="Concatenated code">concatenated</a> with an outer <a href="/wiki/Binary_Golay_code" title="Binary Golay code">Golay (24,12,8) code</a>. The Voyager 2 craft additionally supported an implementation of a <a href="/wiki/Reed%E2%80%93Solomon_code" class="mw-redirect" title="Reed–Solomon code">Reed–Solomon code</a>. The concatenated Reed–Solomon–Viterbi (RSV) code allowed for very powerful error correction, and enabled the spacecraft's extended journey to <a href="/wiki/Uranus" title="Uranus">Uranus</a> and <a href="/wiki/Neptune" title="Neptune">Neptune</a>. After ECC system upgrades in 1989, both crafts used V2 RSV coding. </p><p>The <a href="/wiki/Consultative_Committee_for_Space_Data_Systems" title="Consultative Committee for Space Data Systems">Consultative Committee for Space Data Systems</a> currently recommends usage of error correction codes with performance similar to the Voyager 2 RSV code as a minimum. Concatenated codes are increasingly falling out of favor with space missions, and are replaced by more powerful codes such as <a href="/wiki/Turbo_code" title="Turbo code">Turbo codes</a> or <a href="/wiki/LDPC_code" class="mw-redirect" title="LDPC code">LDPC codes</a>. </p><p>The different kinds of deep space and orbital missions that are conducted suggest that trying to find a one-size-fits-all error correction system will be an ongoing problem. For missions close to Earth, the nature of the <a href="/wiki/Noise_(electronics)" title="Noise (electronics)">noise</a> in the <a href="/wiki/Communication_channel" title="Communication channel">communication channel</a> is different from that which a spacecraft on an interplanetary mission experiences. Additionally, as a spacecraft increases its distance from Earth, the problem of correcting for noise becomes more difficult. </p> <div class="mw-heading mw-heading3"><h3 id="Satellite_broadcasting">Satellite broadcasting</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Error_detection_and_correction&action=edit&section=20" title="Edit section: Satellite broadcasting"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The demand for satellite <a href="/wiki/Transponder" title="Transponder">transponder</a> bandwidth continues to grow, fueled by the desire to deliver television (including new channels and <a href="/wiki/High-definition_television" title="High-definition television">high-definition television</a>) and IP data. Transponder availability and bandwidth constraints have limited this growth. Transponder capacity is determined by the selected <a href="/wiki/Modulation" title="Modulation">modulation</a> scheme and the proportion of capacity consumed by FEC. </p> <div class="mw-heading mw-heading3"><h3 id="Data_storage">Data storage</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Error_detection_and_correction&action=edit&section=21" title="Edit section: Data storage"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Error detection and correction codes are often used to improve the reliability of data storage media.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> A parity track capable of detecting single-bit errors was present on the first <a href="/wiki/Magnetic_tape_data_storage" class="mw-redirect" title="Magnetic tape data storage">magnetic tape data storage</a> in 1951. The <a href="/wiki/Optimal_rectangular_code" class="mw-redirect" title="Optimal rectangular code">optimal rectangular code</a> used in <a href="/wiki/Group_coded_recording" title="Group coded recording">group coded recording</a> tapes not only detects but also corrects single-bit errors. Some <a href="/wiki/File_format" title="File format">file formats</a>, particularly <a href="/wiki/Archive_formats" class="mw-redirect" title="Archive formats">archive formats</a>, include a checksum (most often <a href="/wiki/CRC32" class="mw-redirect" title="CRC32">CRC32</a>) to detect corruption and truncation and can employ redundancy or <a href="/wiki/Parity_file" class="mw-redirect" title="Parity file">parity files</a> to recover portions of corrupted data. <a href="/wiki/Cross-interleaved_Reed%E2%80%93Solomon_coding" title="Cross-interleaved Reed–Solomon coding">Reed-Solomon codes</a> are used in <a href="/wiki/Compact_disc" title="Compact disc">compact discs</a> to correct errors caused by scratches. </p><p>Modern hard drives use Reed–Solomon codes to detect and correct minor errors in sector reads, and to recover corrupted data from failing sectors and store that data in the spare sectors.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> <a href="/wiki/RAID" title="RAID">RAID</a> systems use a variety of error correction techniques to recover data when a hard drive completely fails. Filesystems such as <a href="/wiki/ZFS" title="ZFS">ZFS</a> or <a href="/wiki/Btrfs" title="Btrfs">Btrfs</a>, as well as some <a href="/wiki/RAID" title="RAID">RAID</a> implementations, support <a href="/wiki/Data_scrubbing" title="Data scrubbing">data scrubbing</a> and resilvering, which allows bad blocks to be detected and (hopefully) recovered before they are used.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> The recovered data may be re-written to exactly the same physical location, to spare blocks elsewhere on the same piece of hardware, or the data may be rewritten onto replacement hardware. </p> <div class="mw-heading mw-heading3"><h3 id="Error-correcting_memory"><span class="anchor" id="LINUX-EDAC"></span><span class="anchor" id="BLUESMOKE"></span>Error-correcting memory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Error_detection_and_correction&action=edit&section=22" title="Edit section: Error-correcting memory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/ECC_memory" title="ECC memory">ECC memory</a></div> <p><a href="/wiki/Dynamic_random-access_memory" title="Dynamic random-access memory">Dynamic random-access memory</a> (DRAM) may provide stronger protection against <a href="/wiki/Soft_error" title="Soft error">soft errors</a> by relying on error-correcting codes. Such error-correcting memory, known as <i>ECC</i> or <i>EDAC-protected</i> memory, is particularly desirable for mission-critical applications, such as scientific computing, financial, medical, etc. as well as extraterrestrial applications due to the increased <a href="/wiki/Cosmic_ray" title="Cosmic ray">radiation</a> in space. </p><p>Error-correcting memory controllers traditionally use <a href="/wiki/Hamming_code" title="Hamming code">Hamming codes</a>, although some use <a href="/wiki/Triple_modular_redundancy" title="Triple modular redundancy">triple modular redundancy</a>. <a href="/wiki/Error_correction_code#Interleaving" title="Error correction code">Interleaving</a> allows distributing the effect of a single cosmic ray potentially upsetting multiple physically neighboring bits across multiple words by associating neighboring bits to different words. As long as a <a href="/wiki/Single-event_upset" title="Single-event upset">single-event upset</a> (SEU) does not exceed the error threshold (e.g., a single error) in any particular word between accesses, it can be corrected (e.g., by a single-bit error-correcting code), and the illusion of an error-free memory system may be maintained.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p><p>In addition to hardware providing features required for ECC memory to operate, <a href="/wiki/Operating_system" title="Operating system">operating systems</a> usually contain related reporting facilities that are used to provide notifications when soft errors are transparently recovered. One example is the <a href="/wiki/Linux_kernel" title="Linux kernel">Linux kernel</a>'s <i>EDAC</i> subsystem (previously known as <i>Bluesmoke</i>), which collects the data from error-checking-enabled components inside a computer system; besides collecting and reporting back the events related to ECC memory, it also supports other checksumming errors, including those detected on the <a href="/wiki/PCI_bus" class="mw-redirect" title="PCI bus">PCI bus</a>.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> A few systems<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="Statement needs to be more specific about the content to which it refers. (December 2021)">specify</span></a></i>]</sup> also support <a href="/wiki/Memory_scrubbing" title="Memory scrubbing">memory scrubbing</a> to catch and correct errors early before they become unrecoverable. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Error_detection_and_correction&action=edit&section=23" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Berger_code" title="Berger code">Berger code</a></li> <li><a href="/wiki/Burst_error-correcting_code" title="Burst error-correcting code">Burst error-correcting code</a></li> <li><a href="/wiki/ECC_memory" title="ECC memory">ECC memory</a>, a type of computer data storage</li> <li><a href="/wiki/Link_adaptation" title="Link adaptation">Link adaptation</a></li> <li><a href="/wiki/List_of_algorithms#Error_detection_and_correction" title="List of algorithms">List of algorithms § Error detection and correction</a></li> <li><a href="/wiki/List_of_hash_functions" title="List of hash functions">List of hash functions</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Error_detection_and_correction&action=edit&section=24" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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 mw-references-columns"><ol class="references"> <li id="cite_note-Jewish-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Jewish_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Jewish_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><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 class="citation encyclopaedia cs1">"Masorah". <a rel="nofollow" class="external text" href="http://www.jewishencyclopedia.com/articles/10465-masorah"><i>Jewish Encyclopedia</i></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Masorah&rft.btitle=Jewish+Encyclopedia&rft_id=http%3A%2F%2Fwww.jewishencyclopedia.com%2Farticles%2F10465-masorah&rfr_id=info%3Asid%2Fen.wikipedia.org%3AError+detection+and+correction" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPraticoPelt2009" class="citation book cs1">Pratico, Gary D.; Pelt, Miles V. Van (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=p3FIoT0s3yYC&pg=PT448"><i>Basics of Biblical Hebrew Grammar: Second Edition</i></a>. Zondervan. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-310-55882-8" title="Special:BookSources/978-0-310-55882-8"><bdi>978-0-310-55882-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basics+of+Biblical+Hebrew+Grammar%3A+Second+Edition&rft.pub=Zondervan&rft.date=2009&rft.isbn=978-0-310-55882-8&rft.aulast=Pratico&rft.aufirst=Gary+D.&rft.au=Pelt%2C+Miles+V.+Van&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dp3FIoT0s3yYC%26pg%3DPT448&rfr_id=info%3Asid%2Fen.wikipedia.org%3AError+detection+and+correction" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMounce2007" class="citation book cs1">Mounce, William D. (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=AF-5ptJ0l2gC&pg=PA289"><i>Greek for the Rest of Us: Using Greek Tools Without Mastering Biblical Languages</i></a>. Zondervan. p. 289. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-310-28289-1" title="Special:BookSources/978-0-310-28289-1"><bdi>978-0-310-28289-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Greek+for+the+Rest+of+Us%3A+Using+Greek+Tools+Without+Mastering+Biblical+Languages&rft.pages=289&rft.pub=Zondervan&rft.date=2007&rft.isbn=978-0-310-28289-1&rft.aulast=Mounce&rft.aufirst=William+D.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DAF-5ptJ0l2gC%26pg%3DPA289&rfr_id=info%3Asid%2Fen.wikipedia.org%3AError+detection+and+correction" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a href="/wiki/Mishneh_Torah" title="Mishneh Torah">Mishneh Torah</a>, Tefillin, Mezuzah, and Sefer Torah, 1:2. Example English translation: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEliyahu_Touger" class="citation book cs1"><a href="/w/index.php?title=Eliyahu_Touger&action=edit&redlink=1" class="new" title="Eliyahu Touger (page does not exist)">Eliyahu Touger</a>. <a rel="nofollow" class="external text" href="https://www.chabad.org/library/article_cdo/aid/925417/jewish/Tefillin-Mezuzah-and-Sefer-Torah-Chapter-One.htm"><i>The Rambam's Mishneh Torah</i></a>. <a href="/w/index.php?title=Moznaim_Publishing_Corporation&action=edit&redlink=1" class="new" title="Moznaim Publishing Corporation (page does not exist)">Moznaim Publishing Corporation</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Rambam%27s+Mishneh+Torah&rft.pub=Moznaim+Publishing+Corporation&rft.au=Eliyahu+Touger&rft_id=https%3A%2F%2Fwww.chabad.org%2Flibrary%2Farticle_cdo%2Faid%2F925417%2Fjewish%2FTefillin-Mezuzah-and-Sefer-Torah-Chapter-One.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AError+detection+and+correction" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrian_M._Fagan1996" class="citation book cs1">Brian M. Fagan (5 December 1996). "Dead Sea Scrolls". <i><a href="/w/index.php?title=The_Oxford_Companion_to_Archaeology&action=edit&redlink=1" class="new" title="The Oxford Companion to Archaeology (page does not exist)">The Oxford Companion to Archaeology</a></i>. <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0195076184" title="Special:BookSources/0195076184"><bdi>0195076184</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Dead+Sea+Scrolls&rft.btitle=The+Oxford+Companion+to+Archaeology&rft.pub=Oxford+University+Press&rft.date=1996-12-05&rft.isbn=0195076184&rft.au=Brian+M.+Fagan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AError+detection+and+correction" class="Z3988"></span></span> </li> <li id="cite_note-Thompson-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Thompson_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThompson1983" class="citation cs2">Thompson, Thomas M. (1983), <i>From Error-Correcting Codes through Sphere Packings to Simple Groups</i>, The Carus Mathematical Monographs (#21), The Mathematical Association of America, p. vii, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-88385-023-0" title="Special:BookSources/0-88385-023-0"><bdi>0-88385-023-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=From+Error-Correcting+Codes+through+Sphere+Packings+to+Simple+Groups&rft.series=The+Carus+Mathematical+Monographs+%28%2321%29&rft.pages=vii&rft.pub=The+Mathematical+Association+of+America&rft.date=1983&rft.isbn=0-88385-023-0&rft.aulast=Thompson&rft.aufirst=Thomas+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AError+detection+and+correction" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShannon1948" class="citation cs2">Shannon, C.E. (1948), "A Mathematical Theory of Communication", <i>Bell System Technical Journal</i>, <b>27</b> (3): 379–423, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fj.1538-7305.1948.tb01338.x">10.1002/j.1538-7305.1948.tb01338.x</a>, <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/10338.dmlcz%2F101429">10338.dmlcz/101429</a></span>, <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/9230594">9230594</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bell+System+Technical+Journal&rft.atitle=A+Mathematical+Theory+of+Communication&rft.volume=27&rft.issue=3&rft.pages=379-423&rft.date=1948&rft_id=info%3Ahdl%2F10338.dmlcz%2F101429&rft_id=info%3Apmid%2F9230594&rft_id=info%3Adoi%2F10.1002%2Fj.1538-7305.1948.tb01338.x&rft.aulast=Shannon&rft.aufirst=C.E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AError+detection+and+correction" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGolay1949" class="citation cs2">Golay, Marcel J. E. (1949), "Notes on Digital Coding", <i>Proc.I.R.E. (I.E.E.E.)</i>, <b>37</b>: 657</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proc.I.R.E.+%28I.E.E.E.%29&rft.atitle=Notes+on+Digital+Coding&rft.volume=37&rft.pages=657&rft.date=1949&rft.aulast=Golay&rft.aufirst=Marcel+J.+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AError+detection+and+correction" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGuptaVerma2012" class="citation journal cs1">Gupta, Vikas; Verma, Chanderkant (November 2012). "Error Detection and Correction: An Introduction". <i>International Journal of Advanced Research in Computer Science and Software Engineering</i>. <b>2</b> (11). <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:17499858">17499858</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+Advanced+Research+in+Computer+Science+and+Software+Engineering&rft.atitle=Error+Detection+and+Correction%3A+An+Introduction&rft.volume=2&rft.issue=11&rft.date=2012-11&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A17499858%23id-name%3DS2CID&rft.aulast=Gupta&rft.aufirst=Vikas&rft.au=Verma%2C+Chanderkant&rfr_id=info%3Asid%2Fen.wikipedia.org%3AError+detection+and+correction" class="Z3988"></span></span> </li> <li id="cite_note-reliable-erasure-code-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-reliable-erasure-code_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-reliable-erasure-code_10-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">A. 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McAuley, <i>Reliable Broadband Communication Using a Burst Erasure Correcting Code</i>, ACM SIGCOMM, 1990.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShahVyavahareJain2015" class="citation book cs1">Shah, Pradeep M.; Vyavahare, Prakash D.; Jain, Anjana (September 2015). <a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/301611980">"Modern error correcting codes for 4G and beyond: Turbo codes and LDPC codes"</a>. <i>2015 Radio and Antenna Days of the Indian Ocean (RADIO)</i>. pp. 1–2. <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%2FRADIO.2015.7323369">10.1109/RADIO.2015.7323369</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-9-9903-7339-4" title="Special:BookSources/978-9-9903-7339-4"><bdi>978-9-9903-7339-4</bdi></a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:28885076">28885076</a><span class="reference-accessdate">. Retrieved <span class="nowrap">22 May</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Modern+error+correcting+codes+for+4G+and+beyond%3A+Turbo+codes+and+LDPC+codes&rft.btitle=2015+Radio+and+Antenna+Days+of+the+Indian+Ocean+%28RADIO%29&rft.pages=1-2&rft.date=2015-09&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A28885076%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1109%2FRADIO.2015.7323369&rft.isbn=978-9-9903-7339-4&rft.aulast=Shah&rft.aufirst=Pradeep+M.&rft.au=Vyavahare%2C+Prakash+D.&rft.au=Jain%2C+Anjana&rft_id=https%3A%2F%2Fwww.researchgate.net%2Fpublication%2F301611980&rfr_id=info%3Asid%2Fen.wikipedia.org%3AError+detection+and+correction" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation journal cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20220522205452/https://standards.ieee.org/ieee/802.11ac/4473/">"IEEE SA - IEEE 802.11ac-2013"</a>. <i>IEEE Standards Association</i>. 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Retrieved <span class="nowrap">2022-05-22</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Standards+Association&rft.atitle=IEEE+SA+-+IEEE+802.11ac-2013&rft_id=https%3A%2F%2Fstandards.ieee.org%2Fieee%2F802.11ac%2F4473%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AError+detection+and+correction" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.seagate.com/sg/en/tech-insights/advanced-format-4k-sector-hard-drives-master-ti/">"Transition to Advanced Format 4K Sector Hard Drives | Seagate US"</a>. <i>Seagate.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">22 May</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Seagate.com&rft.atitle=Transition+to+Advanced+Format+4K+Sector+Hard+Drives+%7C+Seagate+US&rft_id=https%3A%2F%2Fwww.seagate.com%2Fsg%2Fen%2Ftech-insights%2Fadvanced-format-4k-sector-hard-drives-master-ti%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AError+detection+and+correction" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrank_van_Gerwen" class="citation web cs1">Frank van Gerwen. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170712053335/http://www.cisquet.nl/numbers.htm">"Numbers (and other mysterious) stations"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.cisquet.nl/numbers.htm">the original</a> on 12 July 2017<span class="reference-accessdate">. Retrieved <span class="nowrap">12 March</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Numbers+%28and+other+mysterious%29+stations&rft.au=Frank+van+Gerwen&rft_id=http%3A%2F%2Fwww.cisquet.nl%2Fnumbers.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AError+detection+and+correction" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGary_Cutlack2010" class="citation web cs1">Gary Cutlack (25 August 2010). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170705131816/https://www.gizmodo.com.au/2010/08/mysterious-russian-numbers-station-changes-broadcast-after-20-years/">"Mysterious Russian 'Numbers Station' Changes Broadcast After 20 Years"</a>. <i>Gizmodo</i>. 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(2003). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20131013171945/http://www.eng.tau.ac.il/~bengal/SCI_paper.pdf">"Self-correcting inspection procedure under inspection errors"</a> <span class="cs1-format">(PDF)</span>. <i>IIE Transactions</i>. IIE Transactions on Quality and Reliability, 34(6), pp. 529-540. Archived from <a rel="nofollow" class="external text" href="http://www.eng.tau.ac.il/~bengal/SCI_paper.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2013-10-13<span class="reference-accessdate">. Retrieved <span class="nowrap">2014-01-10</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IIE+Transactions&rft.atitle=Self-correcting+inspection+procedure+under+inspection+errors&rft.date=2003&rft.au=Ben-Gal+I.&rft.au=Herer+Y.&rft.au=Raz+T.&rft_id=http%3A%2F%2Fwww.eng.tau.ac.il%2F~bengal%2FSCI_paper.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AError+detection+and+correction" class="Z3988"></span></span> </li> <li id="cite_note-deep-space-codes-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-deep-space-codes_17-0">^</a></b></span> <span class="reference-text">K. Andrews et al., <i>The Development of Turbo and LDPC Codes for Deep-Space Applications</i>, Proceedings of the IEEE, Vol. 95, No. 11, Nov. 2007.</span> </li> <li id="cite_note-voyager-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-voyager_18-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHuffmanPless2003" class="citation book cs1">Huffman, William Cary; <a href="/wiki/Vera_Pless" title="Vera Pless">Pless, Vera S.</a> (2003). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/fundamentalsofer0000huff"><i>Fundamentals of Error-Correcting Codes</i></a></span>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-78280-7" title="Special:BookSources/978-0-521-78280-7"><bdi>978-0-521-78280-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Error-Correcting+Codes&rft.pub=Cambridge+University+Press&rft.date=2003&rft.isbn=978-0-521-78280-7&rft.aulast=Huffman&rft.aufirst=William+Cary&rft.au=Pless%2C+Vera+S.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffundamentalsofer0000huff&rfr_id=info%3Asid%2Fen.wikipedia.org%3AError+detection+and+correction" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKurtasVasic2018" class="citation book cs1">Kurtas, Erozan M.; Vasic, Bane (2018-10-03). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Vx_NBQAAQBAJ&q=Error+detection+and+correction+codes+are+often+used+to+improve+the+reliability+of+data+storage+media&pg=PR5"><i>Advanced Error Control Techniques for Data Storage Systems</i></a>. CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4200-3649-7" title="Special:BookSources/978-1-4200-3649-7"><bdi>978-1-4200-3649-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advanced+Error+Control+Techniques+for+Data+Storage+Systems&rft.pub=CRC+Press&rft.date=2018-10-03&rft.isbn=978-1-4200-3649-7&rft.aulast=Kurtas&rft.aufirst=Erozan+M.&rft.au=Vasic%2C+Bane&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DVx_NBQAAQBAJ%26q%3DError%2Bdetection%2Band%2Bcorrection%2Bcodes%2Bare%2Boften%2Bused%2Bto%2Bimprove%2Bthe%2Breliability%2Bof%2Bdata%2Bstorage%2Bmedia%26pg%3DPR5&rfr_id=info%3Asid%2Fen.wikipedia.org%3AError+detection+and+correction" class="Z3988"></span><sup class="noprint Inline-Template"><span style="white-space: nowrap;">[<i><a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot"><span title=" Dead link tagged March 2020">permanent dead link</span></a></i><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span>]</span></sup></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFScott_A._Moulton" class="citation web cs1">Scott A. Moulton. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080202143103/http://www.myharddrivedied.com/presentations_whitepaper.html">"My Hard Drive Died"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.myharddrivedied.com/presentations_whitepaper.html">the original</a> on 2008-02-02.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=My+Hard+Drive+Died&rft.au=Scott+A.+Moulton&rft_id=http%3A%2F%2Fwww.myharddrivedied.com%2Fpresentations_whitepaper.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AError+detection+and+correction" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFQiaoFuChenSettlemyer2019" class="citation book cs1">Qiao, Zhi; Fu, Song; Chen, Hsing-Bung; Settlemyer, Bradley (2019). "Building Reliable High-Performance Storage Systems: An Empirical and Analytical Study". <i>2019 IEEE International Conference on Cluster Computing (CLUSTER)</i>. pp. 1–10. <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%2FCLUSTER.2019.8891006">10.1109/CLUSTER.2019.8891006</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-7281-4734-5" title="Special:BookSources/978-1-7281-4734-5"><bdi>978-1-7281-4734-5</bdi></a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:207951690">207951690</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Building+Reliable+High-Performance+Storage+Systems%3A+An+Empirical+and+Analytical+Study&rft.btitle=2019+IEEE+International+Conference+on+Cluster+Computing+%28CLUSTER%29&rft.pages=1-10&rft.date=2019&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A207951690%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1109%2FCLUSTER.2019.8891006&rft.isbn=978-1-7281-4734-5&rft.aulast=Qiao&rft.aufirst=Zhi&rft.au=Fu%2C+Song&rft.au=Chen%2C+Hsing-Bung&rft.au=Settlemyer%2C+Bradley&rfr_id=info%3Asid%2Fen.wikipedia.org%3AError+detection+and+correction" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20111002152735/http://www.apmcsta.org/File/doc/Conferences/6th%20meeting/Chen%20Zhenyu.doc">"Using StrongArm SA-1110 in the On-Board Computer of Nanosatellite"</a>. Tsinghua Space Center, <a href="/wiki/Tsinghua_University" title="Tsinghua University">Tsinghua University</a>, Beijing. Archived from <a rel="nofollow" class="external text" href="http://www.apmcsta.org/File/doc/Conferences/6th%20meeting/Chen%20Zhenyu.doc">the original</a> on 2011-10-02<span class="reference-accessdate">. Retrieved <span class="nowrap">2009-02-16</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Using+StrongArm+SA-1110+in+the+On-Board+Computer+of+Nanosatellite&rft.pub=Tsinghua+Space+Center%2C+Tsinghua+University%2C+Beijing&rft_id=http%3A%2F%2Fwww.apmcsta.org%2FFile%2Fdoc%2FConferences%2F6th%2520meeting%2FChen%2520Zhenyu.doc&rfr_id=info%3Asid%2Fen.wikipedia.org%3AError+detection+and+correction" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJeff_Layton" class="citation magazine cs1">Jeff Layton. <a rel="nofollow" class="external text" href="http://www.admin-magazine.com/Articles/Monitoring-Memory-Errors">"Error Detection and Correction"</a>. <i><a href="/wiki/Linux_Magazine" title="Linux Magazine">Linux Magazine</a></i><span class="reference-accessdate">. Retrieved <span class="nowrap">2014-08-12</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Linux+Magazine&rft.atitle=Error+Detection+and+Correction&rft.au=Jeff+Layton&rft_id=http%3A%2F%2Fwww.admin-magazine.com%2FArticles%2FMonitoring-Memory-Errors&rfr_id=info%3Asid%2Fen.wikipedia.org%3AError+detection+and+correction" class="Z3988"></span></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://bluesmoke.sourceforge.net/">"EDAC Project"</a>. <i>bluesmoke.sourceforge.net</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2014-08-12</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=bluesmoke.sourceforge.net&rft.atitle=EDAC+Project&rft_id=http%3A%2F%2Fbluesmoke.sourceforge.net%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AError+detection+and+correction" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20090905174616/http://www.kernel.org/doc/Documentation/edac.txt">"Documentation/edac.txt"</a>. <i>Linux kernel documentation</i>. <a href="/wiki/Kernel.org" title="Kernel.org">kernel.org</a>. 2014-06-16. 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Retrieved <span class="nowrap">2014-08-12</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Linux+kernel+documentation&rft.atitle=Documentation%2Fedac.txt&rft.date=2014-06-16&rft_id=https%3A%2F%2Fwww.kernel.org%2Fdoc%2FDocumentation%2Fedac.txt&rfr_id=info%3Asid%2Fen.wikipedia.org%3AError+detection+and+correction" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Error_detection_and_correction&action=edit&section=25" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShu_LinDaniel_J._Costello,_Jr.1983" class="citation book cs1">Shu Lin; Daniel J. Costello, Jr. (1983). <i>Error Control Coding: Fundamentals and Applications</i>. <a href="/wiki/Prentice_Hall" title="Prentice Hall">Prentice Hall</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-283796-X" title="Special:BookSources/0-13-283796-X"><bdi>0-13-283796-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Error+Control+Coding%3A+Fundamentals+and+Applications&rft.pub=Prentice+Hall&rft.date=1983&rft.isbn=0-13-283796-X&rft.au=Shu+Lin&rft.au=Daniel+J.+Costello%2C+Jr.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AError+detection+and+correction" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://pdos.csail.mit.edu/papers/softecc:ddopson-meng/softecc_ddopson-meng.pdf">SoftECC: A System for Software Memory Integrity Checking</a></li> <li><a rel="nofollow" class="external text" href="http://www.fiala.me/pubs/papers/libsdc11.pdf">A Tunable, Software-based DRAM Error Detection and Correction Library for HPC</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20141107074502/http://www.fiala.me/pubs/papers/libsdc11.pdf">Archived</a> 2014-11-07 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="http://www.fiala.me/pubs/papers/sc12-redmpi.pdf">Detection and Correction of Silent Data Corruption for Large-Scale High-Performance Computing</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20141107074511/http://www.fiala.me/pubs/papers/sc12-redmpi.pdf">Archived</a> 2014-11-07 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Error_detection_and_correction&action=edit&section=26" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.inference.phy.cam.ac.uk/mackay/itila/">The on-line textbook: Information Theory, Inference, and Learning Algorithms</a>, by <a href="/wiki/David_J.C._MacKay" class="mw-redirect" title="David J.C. MacKay">David J.C. MacKay</a>, contains chapters on elementary error-correcting codes; on the theoretical limits of error-correction; and on the latest state-of-the-art error-correcting codes, including <a href="/wiki/Low-density_parity-check_code" title="Low-density parity-check code">low-density parity-check codes</a>, <a href="/wiki/Turbo_code" title="Turbo code">turbo codes</a>, and <a href="/wiki/Fountain_codes" class="mw-redirect" title="Fountain codes">fountain codes</a>.</li> <li><a rel="nofollow" class="external text" href="http://www.eccpage.com/">ECC Page</a> - implementations of popular ECC encoding and decoding routines</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&oldid=1250125443#LINUX-EDAC">https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&oldid=1250125443#LINUX-EDAC</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Error_detection_and_correction" title="Category:Error detection and correction">Error detection and correction</a></li><li><a href="/wiki/Category:Computer_errors" title="Category:Computer errors">Computer errors</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:All_articles_with_dead_external_links" title="Category:All articles with dead external links">All articles with dead external links</a></li><li><a href="/wiki/Category:Articles_with_dead_external_links_from_March_2020" title="Category:Articles with dead external links from March 2020">Articles with dead external links from March 2020</a></li><li><a href="/wiki/Category:Articles_with_permanently_dead_external_links" title="Category:Articles with permanently dead external links">Articles with permanently dead external links</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:Articles_needing_additional_references_from_August_2008" title="Category:Articles needing additional references from August 2008">Articles needing additional references from August 2008</a></li><li><a href="/wiki/Category:All_articles_needing_additional_references" title="Category:All articles needing additional references">All articles needing additional references</a></li><li><a href="/wiki/Category:Articles_needing_more_detailed_references" title="Category:Articles needing more detailed references">Articles needing more detailed references</a></li><li><a href="/wiki/Category:Webarchive_template_wayback_links" title="Category:Webarchive template wayback links">Webarchive template wayback links</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 8 October 2024, at 16:17<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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