Locally recoverable code

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data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Definition" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definition</span> </div> </a> <ul id="toc-Definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Optimal_locally_recoverable_codes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Optimal_locally_recoverable_codes"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Optimal locally recoverable codes</span> </div> </a> <ul id="toc-Optimal_locally_recoverable_codes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tamo–Barg_codes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Tamo–Barg_codes"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Tamo–Barg codes</span> </div> </a> <button aria-controls="toc-Tamo–Barg_codes-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 Tamo–Barg codes subsection</span> </button> <ul id="toc-Tamo–Barg_codes-sublist" class="vector-toc-list"> <li id="toc-Tamo–Barg_construction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tamo–Barg_construction"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Tamo–Barg construction</span> </div> </a> <ul id="toc-Tamo–Barg_construction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Parameters_of_Tamo–Barg_codes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Parameters_of_Tamo–Barg_codes"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Parameters of Tamo–Barg codes</span> </div> </a> <ul id="toc-Parameters_of_Tamo–Barg_codes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example_of_Tamo–Barg_construction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example_of_Tamo–Barg_construction"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Example of Tamo–Barg construction</span> </div> </a> <ul id="toc-Example_of_Tamo–Barg_construction-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Locally_recoverable_codes_with_availability" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Locally_recoverable_codes_with_availability"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Locally recoverable codes with availability</span> </div> </a> <ul id="toc-Locally_recoverable_codes_with_availability-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">5</span> <span>References</span> </div> </a> <ul id="toc-References-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" 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.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-Orphan plainlinks metadata ambox ambox-style ambox-Orphan" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Wiki_letter_w.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/6/6c/Wiki_letter_w.svg/40px-Wiki_letter_w.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/6/6c/Wiki_letter_w.svg/60px-Wiki_letter_w.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/6/6c/Wiki_letter_w.svg/80px-Wiki_letter_w.svg.png 2x" data-file-width="44" data-file-height="44" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>is an <a href="/wiki/Wikipedia:Orphan" title="Wikipedia:Orphan">orphan</a>, as no other articles <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Special:WhatLinksHere&target=Locally_recoverable_code&namespace=0">link to it</a></b>. Please <a href="/wiki/Help:Link" title="Help:Link">introduce links</a> to this page from <span class="plainlinks"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Special%3ASearch&redirs=1&search=Locally+recoverable+code&fulltext=Search&ns0=1&title=Special%3ASearch&advanced=1&fulltext=Locally+recoverable+code">related articles</a></span>; try the <a rel="nofollow" class="external text" href="https://edwardbetts.com/find_link?q=Locally_recoverable_code">Find link tool</a> for suggestions. <span class="date-container"><i>(<span class="date">November 2024</span>)</i></span></div></td></tr></tbody></table> <p><b>Locally recoverable codes</b> are a family of <a href="/wiki/Error_correction_code" title="Error correction code">error correction codes</a> that were introduced first by D. S. Papailiopoulos and A. G. Dimakis<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> and have been widely studied in <a href="/wiki/Information_theory" title="Information theory">information theory</a> due to their applications related to distributive and <a href="/wiki/Cloud_storage" title="Cloud storage">cloud storage</a> systems.<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><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><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>An <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [n,k,d,r]_{q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi>d</mi> <mo>,</mo> <mi>r</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [n,k,d,r]_{q}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dc7efb853a6f556a2b325bb73c3517415f2bd10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.255ex; height:3.009ex;" alt="{\displaystyle [n,k,d,r]_{q}}"></span> LRC is an <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [n,k,d]_{q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi>d</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [n,k,d]_{q}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eef76e1d6b74cc535236a6249e75afeea540ee3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.172ex; height:3.009ex;" alt="{\displaystyle [n,k,d]_{q}}"></span> <a href="/wiki/Linear_code" title="Linear code">linear code</a> such that there is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65da883ca3d16b461e46c94777b0d9c4aa010e79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.509ex;" alt="{\displaystyle f_{i}}"></span> that takes as input <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> and a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> other <a href="/wiki/Coordinate_system" title="Coordinate system">coordinates</a> of a codeword <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=(c_{1},\ldots ,c_{n})\in C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=(c_{1},\ldots ,c_{n})\in C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e0280a862fc02f68518ede82ca3ea14f52be2db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.986ex; height:2.843ex;" alt="{\displaystyle c=(c_{1},\ldots ,c_{n})\in C}"></span> different from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01acb7953ba52c2aa44264b5d0f8fd223aa178a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.807ex; height:2.009ex;" alt="{\displaystyle c_{i}}"></span>, and outputs <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01acb7953ba52c2aa44264b5d0f8fd223aa178a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.807ex; height:2.009ex;" alt="{\displaystyle c_{i}}"></span>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Locally_recoverable_code&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div><p> Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> be a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [n,k,d]_{q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi>d</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [n,k,d]_{q}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eef76e1d6b74cc535236a6249e75afeea540ee3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.172ex; height:3.009ex;" alt="{\displaystyle [n,k,d]_{q}}"></span> <a href="/wiki/Linear_code" title="Linear code">linear code</a>. For <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in \{1,\ldots ,n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\in \{1,\ldots ,n\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7898b0be3a01bf31af25ec96239b2a7b96b77055" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.703ex; height:2.843ex;" alt="{\displaystyle i\in \{1,\ldots ,n\}}"></span>, let us denote by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0b6d651eaf432dbf1f106021c8bb499ae83fd1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.848ex; height:2.009ex;" alt="{\displaystyle r_{i}}"></span> the minimum number of other <a href="/wiki/Coordinate_system" title="Coordinate system">coordinates</a> we have to look at to recover an erasure in <a href="/wiki/Coordinate_system" title="Coordinate system">coordinate</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>. The number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0b6d651eaf432dbf1f106021c8bb499ae83fd1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.848ex; height:2.009ex;" alt="{\displaystyle r_{i}}"></span> is said to be the <i>locality of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>-th <a href="/wiki/Coordinate_system" title="Coordinate system">coordinate</a></i> of the code. The <i>locality</i> of the code is defined as </p><div style="text-align: center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r={\textrm {max}}\{r_{i}|i\in \{1,\ldots ,n\}\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>max</mtext> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>i</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r={\textrm {max}}\{r_{i}|i\in \{1,\ldots ,n\}\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0546e78ae437d5b1bbea654dd0355f1f36d806ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.643ex; height:2.843ex;" alt="{\displaystyle r={\textrm {max}}\{r_{i}|i\in \{1,\ldots ,n\}\}.}"></span></div> <p>An <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [n,k,d,r]_{q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi>d</mi> <mo>,</mo> <mi>r</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [n,k,d,r]_{q}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dc7efb853a6f556a2b325bb73c3517415f2bd10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.255ex; height:3.009ex;" alt="{\displaystyle [n,k,d,r]_{q}}"></span> <i>locally recoverable code</i> (LRC) is an <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [n,k,d]_{q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi>d</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [n,k,d]_{q}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eef76e1d6b74cc535236a6249e75afeea540ee3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.172ex; height:3.009ex;" alt="{\displaystyle [n,k,d]_{q}}"></span> <a href="/wiki/Linear_code" title="Linear code">linear code</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C\in \mathbb {F} _{q}^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>∈</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C\in \mathbb {F} _{q}^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03f01de462df84160df40446a4b8d013b0e8a08f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.246ex; height:3.176ex;" alt="{\displaystyle C\in \mathbb {F} _{q}^{n}}"></span> with locality <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>. </p><p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> be an <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [n,k,d]_{q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi>d</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [n,k,d]_{q}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eef76e1d6b74cc535236a6249e75afeea540ee3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.172ex; height:3.009ex;" alt="{\displaystyle [n,k,d]_{q}}"></span>-locally recoverable code. Then an erased component can be recovered linearly,<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> i.e. for every <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in \{1,\ldots ,n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\in \{1,\ldots ,n\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7898b0be3a01bf31af25ec96239b2a7b96b77055" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.703ex; height:2.843ex;" alt="{\displaystyle i\in \{1,\ldots ,n\}}"></span>, the space of <a href="/wiki/Linear_equation" title="Linear equation">linear equations</a> of the code contains <a href="/wiki/Element_(mathematics)" title="Element (mathematics)">elements</a> of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}=f(x_{i_{1}},\ldots ,x_{i_{r}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}=f(x_{i_{1}},\ldots ,x_{i_{r}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9a1f92f274dc8e842ad1c95550b453f45ef8fcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.35ex; height:3.009ex;" alt="{\displaystyle x_{i}=f(x_{i_{1}},\ldots ,x_{i_{r}})}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{j}\neq i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>≠</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i_{j}\neq i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/076bbc693b7170a549f1dd4f3f503b4ccff8a1c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.613ex; height:2.843ex;" alt="{\displaystyle i_{j}\neq i}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Optimal_locally_recoverable_codes">Optimal locally recoverable codes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Locally_recoverable_code&action=edit&section=2" title="Edit section: Optimal locally recoverable codes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Theorem</b><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> Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=(r+1)s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=(r+1)s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf306566323a7e2ff31fc6b5a1c38c0075c72b47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.444ex; height:2.843ex;" alt="{\displaystyle n=(r+1)s}"></span> and let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> be an <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [n,k,d]_{q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi>d</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [n,k,d]_{q}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eef76e1d6b74cc535236a6249e75afeea540ee3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.172ex; height:3.009ex;" alt="{\displaystyle [n,k,d]_{q}}"></span>-locally recoverable code having <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> disjoint locality <a href="/wiki/Disjoint_sets" title="Disjoint sets">sets</a> of size <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bf1b7e1e8ed2801660c299494ed10e6f18f790d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.052ex; height:2.343ex;" alt="{\displaystyle r+1}"></span>. Then </p><div style="text-align: center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\leq n-k-\left\lceil {\frac {k}{r}}\right\rceil +2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>≤</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>−</mo> <mrow> <mo>⌈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>r</mi> </mfrac> </mrow> <mo>⌉</mo> </mrow> <mo>+</mo> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\leq n-k-\left\lceil {\frac {k}{r}}\right\rceil +2.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b2cff99e4c823e37313cb02f090fa263ee5a917" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.009ex; height:6.176ex;" alt="{\displaystyle d\leq n-k-\left\lceil {\frac {k}{r}}\right\rceil +2.}"></span></div><p> An <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [n,k,d,r]_{q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi>d</mi> <mo>,</mo> <mi>r</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [n,k,d,r]_{q}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dc7efb853a6f556a2b325bb73c3517415f2bd10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.255ex; height:3.009ex;" alt="{\displaystyle [n,k,d,r]_{q}}"></span>-LRC <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> is said to be optimal if the minimum <a href="/wiki/Hamming_distance" title="Hamming distance">distance</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> satisfies </p><div style="text-align: center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d=n-k-\left\lceil {\frac {k}{r}}\right\rceil +2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>−</mo> <mrow> <mo>⌈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>r</mi> </mfrac> </mrow> <mo>⌉</mo> </mrow> <mo>+</mo> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d=n-k-\left\lceil {\frac {k}{r}}\right\rceil +2.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3de0db4dca727ad30aa41cac92cd6ede1795a2a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.009ex; height:6.176ex;" alt="{\displaystyle d=n-k-\left\lceil {\frac {k}{r}}\right\rceil +2.}"></span></div> <div class="mw-heading mw-heading2"><h2 id="Tamo–Barg_codes"><span id="Tamo.E2.80.93Barg_codes"></span>Tamo–Barg codes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Locally_recoverable_code&action=edit&section=3" title="Edit section: Tamo–Barg codes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in \mathbb {F} _{q}[x]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in \mathbb {F} _{q}[x]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2744d0660023a87aae89aea3552d39e81cb6851c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.151ex; height:3.009ex;" alt="{\displaystyle f\in \mathbb {F} _{q}[x]}"></span> be a <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> and let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }"></span> be a positive <a href="/wiki/Integer" title="Integer">integer</a>. Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> is said to be (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }"></span>)-good if </p> <dl><dd>• <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> has <a href="/wiki/Degree_of_a_polynomial" title="Degree of a polynomial">degree</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bf1b7e1e8ed2801660c299494ed10e6f18f790d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.052ex; height:2.343ex;" alt="{\displaystyle r+1}"></span>,</dd></dl> <dl><dd>• there exist distinct <a href="/wiki/Subset" title="Subset">subsets</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1},\ldots ,A_{\ell }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1},\ldots ,A_{\ell }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de49fb0a3e0438e6eea8208984b6488db99538c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.637ex; height:2.509ex;" alt="{\displaystyle A_{1},\ldots ,A_{\ell }}"></span> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{q}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbb96e056c071d13fc7702013f9273e7f5cd88a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.409ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} _{q}}"></span> such that</dd></dl> <dl><dd><dl><dd>– for any <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in \{1,\ldots ,\ell \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>ℓ</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\in \{1,\ldots ,\ell \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0944b6fbd3ab232ffeee17bed5a847132afc4575" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.278ex; height:2.843ex;" alt="{\displaystyle i\in \{1,\ldots ,\ell \}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(A_{i})=\{t_{i}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(A_{i})=\{t_{i}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65d7e926fc46d61b39b462b66576064174da1e38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.693ex; height:2.843ex;" alt="{\displaystyle f(A_{i})=\{t_{i}\}}"></span> for some <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{i}\in \mathbb {F} _{q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{i}\in \mathbb {F} _{q}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c8760350012a306d2565e374be4a56fb7f42a87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.889ex; height:2.843ex;" alt="{\displaystyle t_{i}\in \mathbb {F} _{q}}"></span> , i.e., <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> is <a href="/wiki/Constant_(mathematics)" title="Constant (mathematics)">constant</a> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1aed3b5def921afbe6cc48aaf8f9b11c6f1c1e2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.543ex; height:2.509ex;" alt="{\displaystyle A_{i}}"></span>,</dd></dl></dd></dl> <dl><dd><dl><dd>– <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \#A_{i}=r+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">#</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>r</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \#A_{i}=r+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a86acce714912dc60b59a24f617dc969868b01a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.629ex; height:2.509ex;" alt="{\displaystyle \#A_{i}=r+1}"></span>,</dd></dl></dd></dl> <dl><dd><dl><dd>– <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i}\cap A_{j}=\varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∩</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{i}\cap A_{j}=\varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a970bc33f4f140ab7fe3e0bee5435f86ec0cf4a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.685ex; height:2.843ex;" alt="{\displaystyle A_{i}\cap A_{j}=\varnothing }"></span> for any <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\neq j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>≠</mo> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\neq j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d95aeb406bb427ac96806bc00c30c91d31b858be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.859ex; height:2.676ex;" alt="{\displaystyle i\neq j}"></span>.</dd></dl></dd></dl> <p>We say that {<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1},\ldots ,A_{\ell }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1},\ldots ,A_{\ell }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de49fb0a3e0438e6eea8208984b6488db99538c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.637ex; height:2.509ex;" alt="{\displaystyle A_{1},\ldots ,A_{\ell }}"></span>} is a splitting covering for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>.<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-heading3"><h3 id="Tamo–Barg_construction"><span id="Tamo.E2.80.93Barg_construction"></span>Tamo–Barg construction</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Locally_recoverable_code&action=edit&section=4" title="Edit section: Tamo–Barg construction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Tamo–Barg construction utilizes good polynomials.<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> <dl><dd>• Suppose that a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r,\ell )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>ℓ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r,\ell )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8638c0b218440f24dc34234ae9c28f20165db910" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.862ex; height:2.843ex;" alt="{\displaystyle (r,\ell )}"></span>-good polynomial <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{q}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbb96e056c071d13fc7702013f9273e7f5cd88a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.409ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} _{q}}"></span> is given with splitting covering <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in \{1,\ldots ,\ell \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>ℓ</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\in \{1,\ldots ,\ell \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0944b6fbd3ab232ffeee17bed5a847132afc4575" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.278ex; height:2.843ex;" alt="{\displaystyle i\in \{1,\ldots ,\ell \}}"></span>.</dd> <dd>• Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\leq \ell -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>≤</mo> <mi>ℓ</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\leq \ell -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba6ac878a12a37866518036599292a7fe20c31ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.161ex; height:2.343ex;" alt="{\displaystyle s\leq \ell -1}"></span> be a positive <a href="/wiki/Integer" title="Integer">integer</a>.</dd> <dd>• Consider the following <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{q}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbb96e056c071d13fc7702013f9273e7f5cd88a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.409ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} _{q}}"></span>-<a href="/wiki/Vector_space" title="Vector space">vector space</a> of <a href="/wiki/Polynomial" title="Polynomial">polynomials</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=\{\sum _{i=0}^{s}g_{i}(x)f(x)^{i}:\deg(g_{i}(x))\leq \deg(f(x))-2\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </munderover> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>:</mo> <mi>deg</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>deg</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=\{\sum _{i=0}^{s}g_{i}(x)f(x)^{i}:\deg(g_{i}(x))\leq \deg(f(x))-2\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a765ad6ebeaec7e96d78ba63f2967000bdaa41d6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:50.962ex; height:6.843ex;" alt="{\displaystyle V=\{\sum _{i=0}^{s}g_{i}(x)f(x)^{i}:\deg(g_{i}(x))\leq \deg(f(x))-2\}.}"></span></dd> <dd>• Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle T=\bigcup _{i=1}^{\ell }A_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <munderover> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </munderover> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle T=\bigcup _{i=1}^{\ell }A_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3285581251451513635aad6aa87fb2b92b89116" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.501ex; height:3.509ex;" alt="{\textstyle T=\bigcup _{i=1}^{\ell }A_{i}}"></span>.</dd> <dd>• The code <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{ev_{T}(g):g\in V\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>e</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>g</mi> <mo>∈</mo> <mi>V</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{ev_{T}(g):g\in V\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d524ac4180e64ef5a42ab18b1d86a22517d23f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.532ex; height:2.843ex;" alt="{\displaystyle \{ev_{T}(g):g\in V\}}"></span> is an <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((r+1)\ell ,(s+1)r,d,r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>ℓ</mi> <mo>,</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>r</mi> <mo>,</mo> <mi>d</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((r+1)\ell ,(s+1)r,d,r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72339861e98c4ae93d1cbd12fee3c4276eae2b62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.957ex; height:2.843ex;" alt="{\displaystyle ((r+1)\ell ,(s+1)r,d,r)}"></span>-optimal locally coverable code, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ev_{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ev_{T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2aec9b8eb735a1889fa302faaea6edcef6ef243" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.6ex; height:2.009ex;" alt="{\displaystyle ev_{T}}"></span> denotes evaluation of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> at all points in the <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Parameters_of_Tamo–Barg_codes"><span id="Parameters_of_Tamo.E2.80.93Barg_codes"></span>Parameters of Tamo–Barg codes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Locally_recoverable_code&action=edit&section=5" title="Edit section: Parameters of Tamo–Barg codes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd>• <b>Length.</b> The length is the number of evaluation points. Because the <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1aed3b5def921afbe6cc48aaf8f9b11c6f1c1e2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.543ex; height:2.509ex;" alt="{\displaystyle A_{i}}"></span> are <a href="/wiki/Disjoint_sets" title="Disjoint sets">disjoint</a> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in \{1,\ldots ,\ell \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>ℓ</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\in \{1,\ldots ,\ell \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0944b6fbd3ab232ffeee17bed5a847132afc4575" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.278ex; height:2.843ex;" alt="{\displaystyle i\in \{1,\ldots ,\ell \}}"></span>, the length of the code is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |T|=(r+1)\ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |T|=(r+1)\ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eeb2d4817facb01e702506c01baa4c87cda5a208" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.859ex; height:2.843ex;" alt="{\displaystyle |T|=(r+1)\ell }"></span>.</dd></dl> <dl><dd>• <b>Dimension.</b> The <a href="/wiki/Dimension" title="Dimension">dimension</a> of the code is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (s+1)r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (s+1)r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ee48f5cbb9020242def022e704997572e5a8bae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.951ex; height:2.843ex;" alt="{\displaystyle (s+1)r}"></span>, for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> ≤ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8144f37f93f6f504a155fa04a07f0da8917fab1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.973ex; height:2.343ex;" alt="{\displaystyle \ell -1}"></span>, as each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ce36142a0a1c6660e82bdf3ef3f1551317efe0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.909ex; height:2.009ex;" alt="{\displaystyle g_{i}}"></span> has <a href="/wiki/Degree_of_a_polynomial" title="Degree of a polynomial">degree</a> at most <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \deg(f(x))-2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>deg</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \deg(f(x))-2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9adf41a4343bc036dd08ef28cad637148f98f8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.717ex; height:2.843ex;" alt="{\displaystyle \deg(f(x))-2}"></span>, covering a <a href="/wiki/Vector_space" title="Vector space">vector space</a> of <a href="/wiki/Dimension" title="Dimension">dimension</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \deg(f(x))-1=r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>deg</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \deg(f(x))-1=r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33caf14249cfa37b1d06215ac2d5587c0dbd1e10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.864ex; height:2.843ex;" alt="{\displaystyle \deg(f(x))-1=r}"></span>, and by the construction of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>, there are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc3e2e7deb52c40fc144183ba49b8494015640ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.093ex; height:2.343ex;" alt="{\displaystyle s+1}"></span> distinct <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ce36142a0a1c6660e82bdf3ef3f1551317efe0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.909ex; height:2.009ex;" alt="{\displaystyle g_{i}}"></span>.</dd></dl> <dl><dd>• <b>Distance.</b> The <a href="/wiki/Hamming_distance" title="Hamming distance">distance</a> is given by the fact that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\subseteq \mathbb {F} _{q}[x]_{\leq k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>⊆</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>x</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>≤</mo> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\subseteq \mathbb {F} _{q}[x]_{\leq k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e41752968e97415360d5636f1ae7ab3e7ea6c373" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.285ex; height:3.009ex;" alt="{\displaystyle V\subseteq \mathbb {F} _{q}[x]_{\leq k}}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=r+1-2+s(r+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mi>r</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mo>+</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=r+1-2+s(r+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9a90aee875b487766cc0fcfef677788bc36a2a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.156ex; height:2.843ex;" alt="{\displaystyle k=r+1-2+s(r+1)}"></span>, and the obtained code is the <a href="/wiki/Reed%E2%80%93Solomon_error_correction" title="Reed–Solomon error correction">Reed-Solomon code</a> of <a href="/wiki/Degree_of_a_polynomial" title="Degree of a polynomial">degree</a> at most <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>, so the minimum <a href="/wiki/Hamming_distance" title="Hamming distance">distance</a> equals <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r+1)\ell -((r+1)-2+s(r+1))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>r</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>ℓ</mi> <mo>−</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <mo>+</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r+1)\ell -((r+1)-2+s(r+1))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ccc4b43573af51eea913ef097b89f338c89af49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.135ex; height:2.843ex;" alt="{\displaystyle (r+1)\ell -((r+1)-2+s(r+1))}"></span>.</dd></dl> <dl><dd>• <b>Locality.</b> After the erasure of the single component, the evaluation at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}\in A_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}\in A_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9498f2f918d0b59e82c9346aa62925f06ff9fae8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.413ex; height:2.509ex;" alt="{\displaystyle a_{i}\in A_{i}}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |A_{i}|=r+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>r</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |A_{i}|=r+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa3b1d6885c3f48010e69a1bee91ea2ce779f52c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.986ex; height:2.843ex;" alt="{\displaystyle |A_{i}|=r+1}"></span>, is unknown, but the evaluations for all other <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in A_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in A_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0a291c98d6222aa149241b038ef255ca609a6f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.613ex; height:2.509ex;" alt="{\displaystyle a\in A_{i}}"></span> are known, so at most <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> evaluations are needed to uniquely determine the erased component, which gives us the locality of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>.</dd></dl> <dl><dd>To see this, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> restricted to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6019bb70c912e59e9d5f442e9217517743ed4831" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.653ex; height:2.843ex;" alt="{\displaystyle A_{j}}"></span> can be described by a <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> of <a href="/wiki/Degree_of_a_polynomial" title="Degree of a polynomial">degree</a> at most <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \deg(f(x))-2=r+1-2=r-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>deg</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mi>r</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mi>r</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \deg(f(x))-2=r+1-2=r-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ac1494980148eb5c55cdf58c8da0a7dab5d1413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.02ex; height:2.843ex;" alt="{\displaystyle \deg(f(x))-2=r+1-2=r-1}"></span> thanks to the form of the <a href="/wiki/Element_(mathematics)" title="Element (mathematics)">elements</a> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> (i.e., thanks to the fact that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> is <a href="/wiki/Constant_function" title="Constant function">constant</a> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6019bb70c912e59e9d5f442e9217517743ed4831" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.653ex; height:2.843ex;" alt="{\displaystyle A_{j}}"></span>, and the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ce36142a0a1c6660e82bdf3ef3f1551317efe0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.909ex; height:2.009ex;" alt="{\displaystyle g_{i}}"></span>'s have <a href="/wiki/Degree_of_a_polynomial" title="Degree of a polynomial">degree</a> at most <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \deg(f(x))-2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>deg</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \deg(f(x))-2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9adf41a4343bc036dd08ef28cad637148f98f8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.717ex; height:2.843ex;" alt="{\displaystyle \deg(f(x))-2}"></span>). On the other hand <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |A_{j}\backslash \{a_{j}\}|=r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi class="MJX-variant" mathvariant="normal">∖</mi> <mo fence="false" stretchy="false">{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |A_{j}\backslash \{a_{j}\}|=r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bccb9b29359f46c286621f318a48a282a33bda1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.721ex; height:3.009ex;" alt="{\displaystyle |A_{j}\backslash \{a_{j}\}|=r}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> evaluations uniquely determine a <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> of <a href="/wiki/Degree_of_a_polynomial" title="Degree of a polynomial">degree</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ad04896104e929da32fad148e240b3fd8dfa874" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.052ex; height:2.343ex;" alt="{\displaystyle r-1}"></span>. Therefore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> can be constructed and evaluated at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0096fb78d6843c9fb67a840dc796b61ad93eec2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.14ex; height:2.343ex;" alt="{\displaystyle a_{j}}"></span> to recover <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(a_{j})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(a_{j})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62364e19c48261fc33af6fab95fed3c053446b8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.065ex; height:3.009ex;" alt="{\displaystyle g(a_{j})}"></span>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Example_of_Tamo–Barg_construction"><span id="Example_of_Tamo.E2.80.93Barg_construction"></span>Example of Tamo–Barg construction</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Locally_recoverable_code&action=edit&section=6" title="Edit section: Example of Tamo–Barg construction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>We will use <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{5}\in \mathbb {F} _{41}[x]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>41</mn> </mrow> </msub> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{5}\in \mathbb {F} _{41}[x]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66b40ce98ea0fe92cf62a531dc6586f1437c7873" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.144ex; height:3.176ex;" alt="{\displaystyle x^{5}\in \mathbb {F} _{41}[x]}"></span> to construct <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [15,8,6,4]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>15</mn> <mo>,</mo> <mn>8</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>4</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [15,8,6,4]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8153c186e2aa3ba4f0f7836daf3f1580d3710d87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.208ex; height:2.843ex;" alt="{\displaystyle [15,8,6,4]}"></span>-LRC. Notice that the <a href="/wiki/Degree_of_a_polynomial" title="Degree of a polynomial">degree</a> of this <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> is 5, and it is <a href="/wiki/Constant_(mathematics)" title="Constant (mathematics)">constant</a> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1aed3b5def921afbe6cc48aaf8f9b11c6f1c1e2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.543ex; height:2.509ex;" alt="{\displaystyle A_{i}}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in \{1,\ldots ,8\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>8</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\in \{1,\ldots ,8\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d14e2eeadd41cab498745e32e529fd68459cfb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.471ex; height:2.843ex;" alt="{\displaystyle i\in \{1,\ldots ,8\}}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}=\{1,10,16,18,37\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>10</mn> <mo>,</mo> <mn>16</mn> <mo>,</mo> <mn>18</mn> <mo>,</mo> <mn>37</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1}=\{1,10,16,18,37\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0b3b1a9eccc40489f0558a78521957bad258c39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.819ex; height:2.843ex;" alt="{\displaystyle A_{1}=\{1,10,16,18,37\}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{2}=2A_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{2}=2A_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bb37be079e2ba051a638c8bf7f9d1f2d96b5de0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.856ex; height:2.509ex;" alt="{\displaystyle A_{2}=2A_{1}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{3}=3A_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>3</mn> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{3}=3A_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dbef0e6646ae45afbadf9644254c37e5f4fb7af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.856ex; height:2.509ex;" alt="{\displaystyle A_{3}=3A_{1}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{4}=4A_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mn>4</mn> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{4}=4A_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/115ea1bb6f4e62c51753f486d1c02119a92c36b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.856ex; height:2.509ex;" alt="{\displaystyle A_{4}=4A_{1}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{5}=5A_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>=</mo> <mn>5</mn> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{5}=5A_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f758cafdd2f469413e7bae4728b69639110157e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.856ex; height:2.509ex;" alt="{\displaystyle A_{5}=5A_{1}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{6}=6A_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>=</mo> <mn>6</mn> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{6}=6A_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3085500910cc915b4f4c11d29e32ffb564e74801" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.856ex; height:2.509ex;" alt="{\displaystyle A_{6}=6A_{1}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{7}=11A_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> <mo>=</mo> <mn>11</mn> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{7}=11A_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a76aeb1048321fa13b37f68918d3667b251dd1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.018ex; height:2.509ex;" alt="{\displaystyle A_{7}=11A_{1}}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{8}=15A_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> <mo>=</mo> <mn>15</mn> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{8}=15A_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bde4776ef4afd6b784855601b812aa5dc8f3b31e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.018ex; height:2.509ex;" alt="{\displaystyle A_{8}=15A_{1}}"></span>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}^{5}=\{1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1}^{5}=\{1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9b85bf0321cfe79a5d3b75b81ce9a72b3962a36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.383ex; height:3.176ex;" alt="{\displaystyle A_{1}^{5}=\{1\}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{2}^{5}=\{32\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>32</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{2}^{5}=\{32\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2585a742016f3ba7f547193c753ec961559f405" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.546ex; height:3.176ex;" alt="{\displaystyle A_{2}^{5}=\{32\}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{3}^{5}=\{38\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>38</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{3}^{5}=\{38\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/defdd46d166245cce180fb2e3ade17e637ea93e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.546ex; height:3.176ex;" alt="{\displaystyle A_{3}^{5}=\{38\}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{4}^{5}=\{40\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>40</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{4}^{5}=\{40\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed18ee1a12b75f0d7f66d8162712d898773e0a92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.546ex; height:3.176ex;" alt="{\displaystyle A_{4}^{5}=\{40\}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{5}^{5}=\{9\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>9</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{5}^{5}=\{9\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79d45b796de45a350ac5454fb7e881f2b6f2a43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.383ex; height:3.176ex;" alt="{\displaystyle A_{5}^{5}=\{9\}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{6}^{5}=\{27\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>27</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{6}^{5}=\{27\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/175de8423dd111977467ed4dae5809b7c78745b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.546ex; height:3.176ex;" alt="{\displaystyle A_{6}^{5}=\{27\}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{7}^{5}=\{3\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{7}^{5}=\{3\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9f7764dea2aa55478ce40712f80398e78714656" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.383ex; height:3.176ex;" alt="{\displaystyle A_{7}^{5}=\{3\}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{8}^{5}=\{14\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>14</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{8}^{5}=\{14\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd9d339ff522547fe5d6f167755eaa82d96ed922" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.546ex; height:3.176ex;" alt="{\displaystyle A_{8}^{5}=\{14\}}"></span>. Hence, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e07d6b7e67f8ecc28b9c3206f192b4144168b61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" alt="{\displaystyle x^{5}}"></span> is a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (4,8)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>4</mn> <mo>,</mo> <mn>8</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (4,8)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c34b45045069c6301c7aec705b6392d3eb37aea4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (4,8)}"></span>-good polynomial over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{41}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>41</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{41}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6107bfc7b73109409c793ab5d8caa99271af811" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.296ex; height:2.509ex;" alt="{\displaystyle \mathbb {F} _{41}}"></span> by the definition. Now, we will use this <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> to construct a code of <a href="/wiki/Dimension" title="Dimension">dimension</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=8}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=8}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1170deafc5d96c9d76fcd097806d334487cddc1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=8}"></span> and length <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=15}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>15</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=15}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15376579267b277377fc09465f44e0ec1c761891" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.818ex; height:2.176ex;" alt="{\displaystyle n=15}"></span> over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{41}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>41</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{41}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6107bfc7b73109409c793ab5d8caa99271af811" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.296ex; height:2.509ex;" alt="{\displaystyle \mathbb {F} _{41}}"></span>. The locality of this code is 4, which will allow us to recover a single <a href="/wiki/Server_(computing)" title="Server (computing)">server</a> failure by looking at the information contained in at most 4 other <a href="/wiki/Server_(computing)" title="Server (computing)">servers</a>. </p><p>Next, let us define the encoding <a href="/wiki/Polynomial" title="Polynomial">polynomial</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{a}(x)=\sum _{i=0}^{r-1}f_{i}(x)x^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{a}(x)=\sum _{i=0}^{r-1}f_{i}(x)x^{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0138d01ee8ba8fb1c3b67d54e80c95d9421226c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.428ex; height:7.343ex;" alt="{\displaystyle f_{a}(x)=\sum _{i=0}^{r-1}f_{i}(x)x^{i}}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}(x)=\sum _{i=0}^{{\frac {k}{r}}-1}a_{i,j}g(x)^{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>r</mi> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{i}(x)=\sum _{i=0}^{{\frac {k}{r}}-1}a_{i,j}g(x)^{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/326ab12594b5e4b8afa24311b60e1fb503553510" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.524ex; height:8.343ex;" alt="{\displaystyle f_{i}(x)=\sum _{i=0}^{{\frac {k}{r}}-1}a_{i,j}g(x)^{j}}"></span>. So, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{a}(x)=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{a}(x)=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46d07ad34a6d88297f93b7c5a4e3820c9d68b2e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.833ex; height:2.843ex;" alt="{\displaystyle f_{a}(x)=}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0,0}+}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mn>0</mn> </mrow> </msub> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0,0}+}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46971d01639d66fbbcac9fc54a568d525513c823" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.372ex; height:2.676ex;" alt="{\displaystyle a_{0,0}+}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0,1}x^{5}+}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0,1}x^{5}+}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e8ab77bad50c43aaa51d58edb53930ad703c1cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.755ex; height:3.343ex;" alt="{\displaystyle a_{0,1}x^{5}+}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1,0}x+}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>0</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1,0}x+}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17622918ed995192ad52b357ed39c0ef0845eefb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.701ex; height:2.676ex;" alt="{\displaystyle a_{1,0}x+}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1,1}x^{6}+}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1,1}x^{6}+}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8be7b0fde0c5e30f3930d94351769137ae35c20b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.755ex; height:3.343ex;" alt="{\displaystyle a_{1,1}x^{6}+}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{2,0}x^{2}+}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>0</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{2,0}x^{2}+}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef5db8c69b339c237dde5d6843b0a0a4242ee08b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.755ex; height:3.343ex;" alt="{\displaystyle a_{2,0}x^{2}+}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{2,1}x^{7}+}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{2,1}x^{7}+}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee7c4dab1f3e0df69dd592a4f7b0c75701669481" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.755ex; height:3.343ex;" alt="{\displaystyle a_{2,1}x^{7}+}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{3,0}x^{3}+}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>,</mo> <mn>0</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{3,0}x^{3}+}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4400b47dc191ec0efc78a5203cd5fbace8c01737" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.755ex; height:3.343ex;" alt="{\displaystyle a_{3,0}x^{3}+}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{3,1}x^{8}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{3,1}x^{8}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d8e020028acd2f9d093e303a41f4ddb819755fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.947ex; height:3.343ex;" alt="{\displaystyle a_{3,1}x^{8}}"></span>. </p><p>Thus, we can use the obtained encoding <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> if we take our <a href="/wiki/Data" title="Data">data</a> to encode as the <a href="/wiki/Row_and_column_vectors" title="Row and column vectors">row vector</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8471c90b288179a1f6eb4fc06d65bad0b1f43f8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.683ex; height:1.676ex;" alt="{\displaystyle a=}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{0,0},a_{0,1},a_{1,0},a_{1,1},a_{2,0},a_{2,1},a_{3,0},a_{3,1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>,</mo> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{0,0},a_{0,1},a_{1,0},a_{1,1},a_{2,0},a_{2,1},a_{3,0},a_{3,1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d297f927661cd7c79a9b7c5362ac4b613b156b1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:37.554ex; height:3.009ex;" alt="{\displaystyle (a_{0,0},a_{0,1},a_{1,0},a_{1,1},a_{2,0},a_{2,1},a_{3,0},a_{3,1})}"></span>. Encoding the <a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vector</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> to a length 15 message <a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vector</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> by multiplying <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> by the <a href="/wiki/Generator_matrix" title="Generator matrix">generator matrix</a> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G={\begin{pmatrix}1&1&1&1&1&1&1&1&1&1&1&1&1&1&1\\1&1&1&1&1&32&32&32&32&32&38&38&38&38&38\\1&10&16&18&37&2&20&32&33&36&3&7&13&29&30\\1&10&16&18&37&23&25&40&31&4&32&20&2&36&33\\1&18&10&37&16&4&31&40&23&25&9&8&5&21&39\\1&18&10&37&16&5&8&9&39&21&14&17&26&19&6\\1&16&37&10&18&8&5&9&21&39&27&15&24&35&22\\1&16&37&10&18&10&37&1&16&18&1&37&10&18&16\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>32</mn> </mtd> <mtd> <mn>32</mn> </mtd> <mtd> <mn>32</mn> </mtd> <mtd> <mn>32</mn> </mtd> <mtd> <mn>32</mn> </mtd> <mtd> <mn>38</mn> </mtd> <mtd> <mn>38</mn> </mtd> <mtd> <mn>38</mn> </mtd> <mtd> <mn>38</mn> </mtd> <mtd> <mn>38</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>10</mn> </mtd> <mtd> <mn>16</mn> </mtd> <mtd> <mn>18</mn> </mtd> <mtd> <mn>37</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>20</mn> </mtd> <mtd> <mn>32</mn> </mtd> <mtd> <mn>33</mn> </mtd> <mtd> <mn>36</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>7</mn> </mtd> <mtd> <mn>13</mn> </mtd> <mtd> <mn>29</mn> </mtd> <mtd> <mn>30</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>10</mn> </mtd> <mtd> <mn>16</mn> </mtd> <mtd> <mn>18</mn> </mtd> <mtd> <mn>37</mn> </mtd> <mtd> <mn>23</mn> </mtd> <mtd> <mn>25</mn> </mtd> <mtd> <mn>40</mn> </mtd> <mtd> <mn>31</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>32</mn> </mtd> <mtd> <mn>20</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>36</mn> </mtd> <mtd> <mn>33</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>18</mn> </mtd> <mtd> <mn>10</mn> </mtd> <mtd> <mn>37</mn> </mtd> <mtd> <mn>16</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>31</mn> </mtd> <mtd> <mn>40</mn> </mtd> <mtd> <mn>23</mn> </mtd> <mtd> <mn>25</mn> </mtd> <mtd> <mn>9</mn> </mtd> <mtd> <mn>8</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>21</mn> </mtd> <mtd> <mn>39</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>18</mn> </mtd> <mtd> <mn>10</mn> </mtd> <mtd> <mn>37</mn> </mtd> <mtd> <mn>16</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>8</mn> </mtd> <mtd> <mn>9</mn> </mtd> <mtd> <mn>39</mn> </mtd> <mtd> <mn>21</mn> </mtd> <mtd> <mn>14</mn> </mtd> <mtd> <mn>17</mn> </mtd> <mtd> <mn>26</mn> </mtd> <mtd> <mn>19</mn> </mtd> <mtd> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>16</mn> </mtd> <mtd> <mn>37</mn> </mtd> <mtd> <mn>10</mn> </mtd> <mtd> <mn>18</mn> </mtd> <mtd> <mn>8</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>9</mn> </mtd> <mtd> <mn>21</mn> </mtd> <mtd> <mn>39</mn> </mtd> <mtd> <mn>27</mn> </mtd> <mtd> <mn>15</mn> </mtd> <mtd> <mn>24</mn> </mtd> <mtd> <mn>35</mn> </mtd> <mtd> <mn>22</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>16</mn> </mtd> <mtd> <mn>37</mn> </mtd> <mtd> <mn>10</mn> </mtd> <mtd> <mn>18</mn> </mtd> <mtd> <mn>10</mn> </mtd> <mtd> <mn>37</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>16</mn> </mtd> <mtd> <mn>18</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>37</mn> </mtd> <mtd> <mn>10</mn> </mtd> <mtd> <mn>18</mn> </mtd> <mtd> <mn>16</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G={\begin{pmatrix}1&1&1&1&1&1&1&1&1&1&1&1&1&1&1\\1&1&1&1&1&32&32&32&32&32&38&38&38&38&38\\1&10&16&18&37&2&20&32&33&36&3&7&13&29&30\\1&10&16&18&37&23&25&40&31&4&32&20&2&36&33\\1&18&10&37&16&4&31&40&23&25&9&8&5&21&39\\1&18&10&37&16&5&8&9&39&21&14&17&26&19&6\\1&16&37&10&18&8&5&9&21&39&27&15&24&35&22\\1&16&37&10&18&10&37&1&16&18&1&37&10&18&16\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afd1be30d2831bbe7dc465323198e803c05ff3ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.171ex; width:76.618ex; height:25.509ex;" alt="{\displaystyle G={\begin{pmatrix}1&1&1&1&1&1&1&1&1&1&1&1&1&1&1\\1&1&1&1&1&32&32&32&32&32&38&38&38&38&38\\1&10&16&18&37&2&20&32&33&36&3&7&13&29&30\\1&10&16&18&37&23&25&40&31&4&32&20&2&36&33\\1&18&10&37&16&4&31&40&23&25&9&8&5&21&39\\1&18&10&37&16&5&8&9&39&21&14&17&26&19&6\\1&16&37&10&18&8&5&9&21&39&27&15&24&35&22\\1&16&37&10&18&10&37&1&16&18&1&37&10&18&16\end{pmatrix}}.}"></span> </p><p>For example, the encoding of information <a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vector</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=(1,1,1,1,1,1,1,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=(1,1,1,1,1,1,1,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47a3f040f2765026dcdb8aa52f793f428aaa260d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.485ex; height:2.843ex;" alt="{\displaystyle m=(1,1,1,1,1,1,1,1)}"></span> gives the codeword <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=mG=(8,8,5,9,21,3,36,31,32,12,2,20,37,33,21)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mi>m</mi> <mi>G</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>8</mn> <mo>,</mo> <mn>8</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>9</mn> <mo>,</mo> <mn>21</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>36</mn> <mo>,</mo> <mn>31</mn> <mo>,</mo> <mn>32</mn> <mo>,</mo> <mn>12</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>20</mn> <mo>,</mo> <mn>37</mn> <mo>,</mo> <mn>33</mn> <mo>,</mo> <mn>21</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=mG=(8,8,5,9,21,3,36,31,32,12,2,20,37,33,21)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/418183828390aeebd328057e6cccf82b44cebc25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.254ex; height:2.843ex;" alt="{\displaystyle c=mG=(8,8,5,9,21,3,36,31,32,12,2,20,37,33,21)}"></span>. </p><p>Observe that we constructed an optimal LRC; therefore, using the <a href="/wiki/Singleton_bound" title="Singleton bound">Singleton bound</a>, we have that the <a href="/wiki/Hamming_distance" title="Hamming distance">distance</a> of this code is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d=n-k-\left\lceil {\frac {k}{r}}\right\rceil +2=15-8-2+2=7}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>−</mo> <mrow> <mo>⌈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>r</mi> </mfrac> </mrow> <mo>⌉</mo> </mrow> <mo>+</mo> <mn>2</mn> <mo>=</mo> <mn>15</mn> <mo>−</mo> <mn>8</mn> <mo>−</mo> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mo>=</mo> <mn>7</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d=n-k-\left\lceil {\frac {k}{r}}\right\rceil +2=15-8-2+2=7}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/deab1ccefd19762d54bf7c8afbf635d4957d1607" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.055ex; height:6.176ex;" alt="{\displaystyle d=n-k-\left\lceil {\frac {k}{r}}\right\rceil +2=15-8-2+2=7}"></span>. Thus, we can recover any 6 erasures from our codeword by looking at no more than 8 other components. </p> <div class="mw-heading mw-heading2"><h2 id="Locally_recoverable_codes_with_availability">Locally recoverable codes with availability</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Locally_recoverable_code&action=edit&section=7" title="Edit section: Locally recoverable codes with availability"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A code <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> has all-symbol locality <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> and availability <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> if every code symbol can be recovered from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> disjoint repair sets of other symbols, each <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of <a href="/wiki/Cardinality" title="Cardinality">size</a> at most <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> symbols. Such codes are called <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r,t)_{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>t</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r,t)_{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c80cb006f630d1af28daf2db5dd83fd47fba76c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.833ex; height:2.843ex;" alt="{\displaystyle (r,t)_{a}}"></span>-LRC.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p><b>Theorem</b> The minimum <a href="/wiki/Hamming_distance" title="Hamming distance">distance</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [n,k,d]_{q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi>d</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [n,k,d]_{q}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eef76e1d6b74cc535236a6249e75afeea540ee3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.172ex; height:3.009ex;" alt="{\displaystyle [n,k,d]_{q}}"></span>-LRC having locality <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> and availability <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> satisfies the <a href="/wiki/Upper_and_lower_bounds" title="Upper and lower bounds">upper bound</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\leq n-\sum _{i=0}^{t}\left\lfloor {\frac {k-1}{r^{i}}}\right\rfloor .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>≤</mo> <mi>n</mi> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </munderover> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mfrac> </mrow> <mo>⌋</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\leq n-\sum _{i=0}^{t}\left\lfloor {\frac {k-1}{r^{i}}}\right\rfloor .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61e057434b351a5356b936564b12b333f2c354a3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.086ex; height:7.176ex;" alt="{\displaystyle d\leq n-\sum _{i=0}^{t}\left\lfloor {\frac {k-1}{r^{i}}}\right\rfloor .}"></span> If the code is <a href="/wiki/Systematic_code" title="Systematic code">systematic</a> and locality and availability apply only to its information symbols, then the code has information locality <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> and availability <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>, and is called <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r,t)_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>t</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r,t)_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47d303df46de33e2cda0dd8d43f6d4ca8e7d58f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.531ex; height:2.843ex;" alt="{\displaystyle (r,t)_{i}}"></span>-LRC.<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> </p><p><b>Theorem</b><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> The minimum <a href="/wiki/Hamming_distance" title="Hamming distance">distance</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> of an <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [n,k,d]_{q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi>d</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [n,k,d]_{q}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eef76e1d6b74cc535236a6249e75afeea540ee3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.172ex; height:3.009ex;" alt="{\displaystyle [n,k,d]_{q}}"></span> linear <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r,t)_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>t</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r,t)_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47d303df46de33e2cda0dd8d43f6d4ca8e7d58f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.531ex; height:2.843ex;" alt="{\displaystyle (r,t)_{i}}"></span>-LRC satisfies the <a href="/wiki/Upper_and_lower_bounds" title="Upper and lower bounds">upper bound</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\leq n-k-\left\lceil {\frac {t(k-1)+1}{t(r-1)+1}}\right\rceil +2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>≤</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>−</mo> <mrow> <mo>⌈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>t</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mi>t</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>⌉</mo> </mrow> <mo>+</mo> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\leq n-k-\left\lceil {\frac {t(k-1)+1}{t(r-1)+1}}\right\rceil +2.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b3404e8c94be7f889a7c2706db9748d71701fe9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:32.663ex; height:6.509ex;" alt="{\displaystyle d\leq n-k-\left\lceil {\frac {t(k-1)+1}{t(r-1)+1}}\right\rceil +2.}"></span> </p> <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=Locally_recoverable_code&action=edit&section=8" 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-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></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 id="CITEREFPapailiopoulosDimakis2012" class="citation cs2">Papailiopoulos, Dimitris S.; 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(2022), "Optimal selection for good polynomials of degree up to five", <i>Designs, Codes and Cryptography</i>, <b>90</b> (6), IEEE: 1427–1436, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10623-022-01046-y">10.1007/s10623-022-01046-y</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Designs%2C+Codes+and+Cryptography&rft.atitle=Optimal+selection+for+good+polynomials+of+degree+up+to+five&rft.volume=90&rft.issue=6&rft.pages=1427-1436&rft.date=2022&rft_id=info%3Adoi%2F10.1007%2Fs10623-022-01046-y&rft.aulast=Dukes&rft.aufirst=A.&rft.au=Ferraguti%2C+A.&rft.au=Micheli%2C+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALocally+recoverable+code" 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="CITEREFHaymakerMalmskogMatthews2022" class="citation cs2">Haymaker, K.; Malmskog, B.; Matthews, G. 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(2012), "Locally repairable codes", <i>2012 IEEE International Symposium on Information Theory</i>, Cambridge, MA, USA, pp. 2771–2775, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1206.3804">1206.3804</a></span>, <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%2FISIT.2012.6284027">10.1109/ISIT.2012.6284027</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4673-2579-0" title="Special:BookSources/978-1-4673-2579-0"><bdi>978-1-4673-2579-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Locally+repairable+codes&rft.btitle=2012+IEEE+International+Symposium+on+Information+Theory&rft.place=Cambridge%2C+MA%2C+USA&rft.pages=2771-2775&rft.date=2012&rft_id=info%3Aarxiv%2F1206.3804&rft_id=info%3Adoi%2F10.1109%2FISIT.2012.6284027&rft.isbn=978-1-4673-2579-0&rft.aulast=Papailiopoulos&rft.aufirst=Dimitris+S.&rft.au=Dimakis%2C+Alexandros+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALocally+recoverable+code" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Citation" title="Template:Citation">citation</a>}}</code>: CS1 maint: location missing publisher (<a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">link</a>)</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="CITEREFCadambeMazumdar2013" class="citation cs2">Cadambe, V.; Mazumdar, A. (2013), "An upper bound on the size of locally recoverable codes", <i>2013 International Symposium on Network Coding</i>, Calgary, AB, Canada, pp. 1–5, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1308.3200">1308.3200</a></span>, <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%2FNetCod.2013.6570829">10.1109/NetCod.2013.6570829</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4799-0823-3" title="Special:BookSources/978-1-4799-0823-3"><bdi>978-1-4799-0823-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=An+upper+bound+on+the+size+of+locally+recoverable+codes&rft.btitle=2013+International+Symposium+on+Network+Coding&rft.place=Calgary%2C+AB%2C+Canada&rft.pages=1-5&rft.date=2013&rft_id=info%3Aarxiv%2F1308.3200&rft_id=info%3Adoi%2F10.1109%2FNetCod.2013.6570829&rft.isbn=978-1-4799-0823-3&rft.aulast=Cadambe&rft.aufirst=V.&rft.au=Mazumdar%2C+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALocally+recoverable+code" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Citation" title="Template:Citation">citation</a>}}</code>: CS1 maint: location missing publisher (<a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">link</a>)</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="CITEREFMicheli2020" class="citation cs2">Micheli, G. (2020), "Constructions of Locally Recoverable Codes Which are Optimal", <i>IEEE Transactions on Information Theory</i>, <b>66</b>: 167–175, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1806.11492">1806.11492</a></span>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTIT.2019.2939464">10.1109/TIT.2019.2939464</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Transactions+on+Information+Theory&rft.atitle=Constructions+of+Locally+Recoverable+Codes+Which+are+Optimal&rft.volume=66&rft.pages=167-175&rft.date=2020&rft_id=info%3Aarxiv%2F1806.11492&rft_id=info%3Adoi%2F10.1109%2FTIT.2019.2939464&rft.aulast=Micheli&rft.aufirst=G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALocally+recoverable+code" 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="CITEREFTamoBarg2014" class="citation cs2">Tamo, I.; Barg, A. (2014), "A family of optimal locally recoverable codes", <i>2014 IEEE International Symposium on Information Theory</i>, Honolulu, HI, USA, pp. 686–690, <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%2FISIT.2014.6874920">10.1109/ISIT.2014.6874920</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4799-5186-4" title="Special:BookSources/978-1-4799-5186-4"><bdi>978-1-4799-5186-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=A+family+of+optimal+locally+recoverable+codes&rft.btitle=2014+IEEE+International+Symposium+on+Information+Theory&rft.place=Honolulu%2C+HI%2C+USA&rft.pages=686-690&rft.date=2014&rft_id=info%3Adoi%2F10.1109%2FISIT.2014.6874920&rft.isbn=978-1-4799-5186-4&rft.aulast=Tamo&rft.aufirst=I.&rft.au=Barg%2C+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALocally+recoverable+code" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Citation" title="Template:Citation">citation</a>}}</code>: CS1 maint: location missing publisher (<a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">link</a>)</span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHuangYaakobiUchikawaSiegel2015" class="citation cs2">Huang, P.; Yaakobi, E.; Uchikawa, H.; Siegel, P.H. (2015), "Linear locally repairable codes with availability", <i>2015 IEEE International Symposium on Information Theory</i>, Hong Kong, China, pp. 1871–1875, <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%2FISIT.2015.7282780">10.1109/ISIT.2015.7282780</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4673-7704-1" title="Special:BookSources/978-1-4673-7704-1"><bdi>978-1-4673-7704-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Linear+locally+repairable+codes+with+availability&rft.btitle=2015+IEEE+International+Symposium+on+Information+Theory&rft.place=Hong+Kong%2C+China&rft.pages=1871-1875&rft.date=2015&rft_id=info%3Adoi%2F10.1109%2FISIT.2015.7282780&rft.isbn=978-1-4673-7704-1&rft.aulast=Huang&rft.aufirst=P.&rft.au=Yaakobi%2C+E.&rft.au=Uchikawa%2C+H.&rft.au=Siegel%2C+P.H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALocally+recoverable+code" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Citation" title="Template:Citation">citation</a>}}</code>: CS1 maint: location missing publisher (<a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">link</a>)</span></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="CITEREFTamoBarg2014" class="citation cs2">Tamo, I.; Barg, A. (2014), "Bounds on locally recoverable codes with multiple recovering sets", <i>2014 IEEE International Symposium on Information Theory</i>, Honolulu, HI, USA, pp. 691–695, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1402.0916">1402.0916</a></span>, <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%2FISIT.2014.6874921">10.1109/ISIT.2014.6874921</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4799-5186-4" title="Special:BookSources/978-1-4799-5186-4"><bdi>978-1-4799-5186-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Bounds+on+locally+recoverable+codes+with+multiple+recovering+sets&rft.btitle=2014+IEEE+International+Symposium+on+Information+Theory&rft.place=Honolulu%2C+HI%2C+USA&rft.pages=691-695&rft.date=2014&rft_id=info%3Aarxiv%2F1402.0916&rft_id=info%3Adoi%2F10.1109%2FISIT.2014.6874921&rft.isbn=978-1-4799-5186-4&rft.aulast=Tamo&rft.aufirst=I.&rft.au=Barg%2C+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALocally+recoverable+code" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Citation" title="Template:Citation">citation</a>}}</code>: CS1 maint: location missing publisher (<a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">link</a>)</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 id="CITEREFWangZhang2014" class="citation cs2">Wang, A.; Zhang, Z. (2014), "Repair locality with multiple erasure tolerance", <i>IEEE Transactions on Information Theory</i>, <b>60</b> (11): 6979–6987, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1306.4774">1306.4774</a></span>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTIT.2014.2351404">10.1109/TIT.2014.2351404</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Transactions+on+Information+Theory&rft.atitle=Repair+locality+with+multiple+erasure+tolerance&rft.volume=60&rft.issue=11&rft.pages=6979-6987&rft.date=2014&rft_id=info%3Aarxiv%2F1306.4774&rft_id=info%3Adoi%2F10.1109%2FTIT.2014.2351404&rft.aulast=Wang&rft.aufirst=A.&rft.au=Zhang%2C+Z.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALocally+recoverable+code" class="Z3988"></span></span> </li> </ol></div></div>    </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=Locally_recoverable_code&oldid=1258174465">https://en.wikipedia.org/w/index.php?title=Locally_recoverable_code&oldid=1258174465</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:Mathematics" title="Category:Mathematics">Mathematics</a></li><li><a href="/wiki/Category:Cryptography" title="Category:Cryptography">Cryptography</a></li><li><a href="/wiki/Category:Information_theory" title="Category:Information theory">Information theory</a></li><li><a href="/wiki/Category:Error_detection_and_correction" title="Category:Error detection and correction">Error detection and correction</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:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">CS1 maint: location missing publisher</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_with_empty_Wikidata_description" title="Category:Short description with empty Wikidata description">Short description with empty Wikidata description</a></li><li><a href="/wiki/Category:Orphaned_articles_from_November_2024" title="Category:Orphaned articles from November 2024">Orphaned articles from November 2024</a></li><li><a href="/wiki/Category:All_orphaned_articles" title="Category:All orphaned articles">All orphaned articles</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 18 November 2024, at 14:56<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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