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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Locally recoverable code</span></span>
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<p><b>Locally recoverable codes</b> are a family of <a href="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="Information_theory" title="Information theory">information theory</a> due to their applications related to distributive and <a href="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 mwe-math-element-inline"><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="./5dc7efb853a6f556a2b325bb73c3517415f2bd10.svg" 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}}" loading="lazy"></span> LRC is an <span class="mwe-math-element mwe-math-element-inline"><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="./5eef76e1d6b74cc535236a6249e75afeea540ee3.svg" 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}}" loading="lazy"></span> <a href="Linear_code" title="Linear code">linear code</a> such that there is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> <span class="mwe-math-element mwe-math-element-inline"><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="./65da883ca3d16b461e46c94777b0d9c4aa010e79.svg" 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}}" loading="lazy"></span> that takes as input <span class="mwe-math-element mwe-math-element-inline"><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="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" 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}" loading="lazy"></span> and a <a href="Set_(mathematics)" title="Set (mathematics)">set</a> of <span class="mwe-math-element mwe-math-element-inline"><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="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" 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}" loading="lazy"></span> other <a href="Coordinate_system" title="Coordinate system">coordinates</a> of a codeword <span class="mwe-math-element mwe-math-element-inline"><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="./5e0280a862fc02f68518ede82ca3ea14f52be2db.svg" 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}" loading="lazy"></span> different from <span class="mwe-math-element mwe-math-element-inline"><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="./01acb7953ba52c2aa44264b5d0f8fd223aa178a2.svg" 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}}" loading="lazy"></span>, and outputs <span class="mwe-math-element mwe-math-element-inline"><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="./01acb7953ba52c2aa44264b5d0f8fd223aa178a2.svg" 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}}" loading="lazy"></span>.
</p>
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<div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2></div>
<p><a href="Error_correction_code" title="Error correction code">Erasure-correcting codes</a>, or simply <a href="Error_correction_code" title="Error correction code">erasure codes</a>, for distributed and <a href="Cloud_storage" title="Cloud storage">cloud storage</a> systems, are becoming more and more popular as a result of the present spike in demand for <a href="Cloud_computing" title="Cloud computing">cloud computing</a> and storage services. This has inspired researchers in the fields of <a href="Information_theory" title="Information theory">information</a> and <a href="Coding_theory" title="Coding theory">coding theory</a> to investigate new facets of codes that are specifically suited for use with storage systems.
</p><p>It is well-known that LRC is a <a href="Linear_code" title="Linear code">code</a> that needs only a limited <a href="Set_(mathematics)" title="Set (mathematics)">set</a> of other symbols to be accessed in order to restore every symbol in a codeword. This idea is very important for distributed and <a href="Cloud_storage" title="Cloud storage">cloud storage</a> systems since the most common error case is when one storage node fails (erasure). The main objective is to recover as much <a href="Data" title="Data">data</a> as possible from the fewest additional storage nodes in order to restore the node. Hence, Locally Recoverable Codes are crucial for such systems.
</p><p>The following <a href="Definition" title="Definition">definition</a> of the LRC follows from the description above: an <span class="mwe-math-element mwe-math-element-inline"><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,r]}">
<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>r</mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [n,k,r]}</annotation>
</semantics>
</math></span><img src="./f90cb467594b0d6ba714b24b7eb83f34d5de125f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.016ex; height:2.843ex;" alt="{\displaystyle [n,k,r]}" loading="lazy"></span>-Locally Recoverable Code (LRC) of length <span class="mwe-math-element mwe-math-element-inline"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> is a <a href="Linear_code" title="Linear code">code</a> that produces an <span class="mwe-math-element mwe-math-element-inline"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>-symbol codeword from <span class="mwe-math-element mwe-math-element-inline"><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="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" 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}" loading="lazy"></span> information symbols, and for any symbol of the codeword, there exist at most <span class="mwe-math-element mwe-math-element-inline"><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="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" 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}" loading="lazy"></span> other symbols such that the value of the symbol can be recovered from them. The locality <a href="Parameter" title="Parameter">parameter</a> satisfies <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq r\leq k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>1</mn>
<mo>≤<!-- ≤ --></mo>
<mi>r</mi>
<mo>≤<!-- ≤ --></mo>
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 1\leq r\leq k}</annotation>
</semantics>
</math></span><img src="./b0aa3854cb060eefd3aacf6bdfa2f7c6f2978c39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.619ex; height:2.343ex;" alt="{\displaystyle 1\leq r\leq k}" loading="lazy"></span> because the entire codeword can be found by accessing <span class="mwe-math-element mwe-math-element-inline"><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="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" 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}" loading="lazy"></span> symbols other than the erased symbol. Furthermore, Locally Recoverable Codes, having the minimum <a href="Hamming_distance" title="Hamming distance">distance</a> <span class="mwe-math-element mwe-math-element-inline"><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="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" 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}" loading="lazy"></span>, can recover <span class="mwe-math-element mwe-math-element-inline"><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-1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>d</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle d-1}</annotation>
</semantics>
</math></span><img src="./0195b64ba44bcc80b4c98e9d34256b4043fe519e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.219ex; height:2.343ex;" alt="{\displaystyle d-1}" loading="lazy"></span> erasures.
</p>
<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div><p>
Let <span class="mwe-math-element mwe-math-element-inline"><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="./4fc55753007cd3c18576f7933f6f089196732029.svg" 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}" loading="lazy"></span> be a <span class="mwe-math-element mwe-math-element-inline"><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="./5eef76e1d6b74cc535236a6249e75afeea540ee3.svg" 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}}" loading="lazy"></span> <a href="Linear_code" title="Linear code">linear code</a>. For <span class="mwe-math-element mwe-math-element-inline"><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="./7898b0be3a01bf31af25ec96239b2a7b96b77055.svg" 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\}}" loading="lazy"></span>, let us denote by <span class="mwe-math-element mwe-math-element-inline"><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="./a0b6d651eaf432dbf1f106021c8bb499ae83fd1f.svg" 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}}" loading="lazy"></span> the minimum number of other <a href="Coordinate_system" title="Coordinate system">coordinates</a> we have to look at to recover an erasure in <a href="Coordinate_system" title="Coordinate system">coordinate</a> <span class="mwe-math-element mwe-math-element-inline"><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="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" 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}" loading="lazy"></span>. The number <span class="mwe-math-element mwe-math-element-inline"><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="./a0b6d651eaf432dbf1f106021c8bb499ae83fd1f.svg" 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}}" loading="lazy"></span> is said to be the <i>locality of the <span class="mwe-math-element mwe-math-element-inline"><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="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" 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}" loading="lazy"></span>-th <a href="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 mwe-math-element-inline"><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=\max\{r_{i}\mid i\in \{1,\ldots ,n\}\}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>r</mi>
<mo>=</mo>
<mo movablelimits="true" form="prefix">max</mo>
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>∣<!-- ∣ --></mo>
<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=\max\{r_{i}\mid i\in \{1,\ldots ,n\}\}.}</annotation>
</semantics>
</math></span><img src="./b699a0dea629ee201877a2723332da0766590f76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.934ex; height:2.843ex;" alt="{\displaystyle r=\max\{r_{i}\mid i\in \{1,\ldots ,n\}\}.}" loading="lazy"></span></div>
<p>An <span class="mwe-math-element mwe-math-element-inline"><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="./5dc7efb853a6f556a2b325bb73c3517415f2bd10.svg" 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}}" loading="lazy"></span> <i>locally recoverable code</i> (LRC) is an <span class="mwe-math-element mwe-math-element-inline"><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="./5eef76e1d6b74cc535236a6249e75afeea540ee3.svg" 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}}" loading="lazy"></span> <a href="Linear_code" title="Linear code">linear code</a> <span class="mwe-math-element mwe-math-element-inline"><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="./03f01de462df84160df40446a4b8d013b0e8a08f.svg" 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}}" loading="lazy"></span> with locality <span class="mwe-math-element mwe-math-element-inline"><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="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" 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}" loading="lazy"></span>.
</p><p>Let <span class="mwe-math-element mwe-math-element-inline"><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="./4fc55753007cd3c18576f7933f6f089196732029.svg" 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}" loading="lazy"></span> be an <span class="mwe-math-element mwe-math-element-inline"><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="./5eef76e1d6b74cc535236a6249e75afeea540ee3.svg" 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}}" loading="lazy"></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 mwe-math-element-inline"><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="./7898b0be3a01bf31af25ec96239b2a7b96b77055.svg" 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\}}" loading="lazy"></span>, the space of <a href="Linear_equation" title="Linear equation">linear equations</a> of the code contains <a href="Element_(mathematics)" title="Element (mathematics)">elements</a> of the form <span class="mwe-math-element mwe-math-element-inline"><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="./e9a1f92f274dc8e842ad1c95550b453f45ef8fcc.svg" 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}})}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><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="./076bbc693b7170a549f1dd4f3f503b4ccff8a1c9.svg" 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}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Optimal_locally_recoverable_codes">Optimal locally recoverable codes</h2></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 mwe-math-element-inline"><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="./cf306566323a7e2ff31fc6b5a1c38c0075c72b47.svg" 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}" loading="lazy"></span> and let <span class="mwe-math-element mwe-math-element-inline"><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="./4fc55753007cd3c18576f7933f6f089196732029.svg" 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}" loading="lazy"></span> be an <span class="mwe-math-element mwe-math-element-inline"><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="./5eef76e1d6b74cc535236a6249e75afeea540ee3.svg" 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}}" loading="lazy"></span>-locally recoverable code having <span class="mwe-math-element mwe-math-element-inline"><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="./01d131dfd7673938b947072a13a9744fe997e632.svg" 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}" loading="lazy"></span> disjoint locality <a href="Disjoint_sets" title="Disjoint sets">sets</a> of size <span class="mwe-math-element mwe-math-element-inline"><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="./7bf1b7e1e8ed2801660c299494ed10e6f18f790d.svg" 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}" loading="lazy"></span>. Then </p><div style="text-align: center;"><span class="mwe-math-element mwe-math-element-inline"><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="./7b2cff99e4c823e37313cb02f090fa263ee5a917.svg" 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.}" loading="lazy"></span></div><p>
An <span class="mwe-math-element mwe-math-element-inline"><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="./5dc7efb853a6f556a2b325bb73c3517415f2bd10.svg" 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}}" loading="lazy"></span>-LRC <span class="mwe-math-element mwe-math-element-inline"><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="./4fc55753007cd3c18576f7933f6f089196732029.svg" 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}" loading="lazy"></span> is said to be optimal if the minimum <a href="Hamming_distance" title="Hamming distance">distance</a> of <span class="mwe-math-element mwe-math-element-inline"><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="./4fc55753007cd3c18576f7933f6f089196732029.svg" 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}" loading="lazy"></span> satisfies </p><div style="text-align: center;"><span class="mwe-math-element mwe-math-element-inline"><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="./3de0db4dca727ad30aa41cac92cd6ede1795a2a7.svg" 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.}" loading="lazy"></span></div>
<div class="mw-heading mw-heading2"><h2 id="Tamo–Barg_codes">Tamo–Barg codes</h2></div>
<p>Let <span class="mwe-math-element mwe-math-element-inline"><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="./2744d0660023a87aae89aea3552d39e81cb6851c.svg" 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]}" loading="lazy"></span> be a <a href="Polynomial" title="Polynomial">polynomial</a> and let <span class="mwe-math-element mwe-math-element-inline"><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<!-- ℓ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \ell }</annotation>
</semantics>
</math></span><img src="./f066e981e530bacc07efc6a10fa82deee985929e.svg" 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 }" loading="lazy"></span> be a positive <a href="Integer" title="Integer">integer</a>. Then <span class="mwe-math-element mwe-math-element-inline"><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="./132e57acb643253e7810ee9702d9581f159a1c61.svg" 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}" loading="lazy"></span> is said to be (<span class="mwe-math-element mwe-math-element-inline"><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="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" 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}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><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<!-- ℓ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \ell }</annotation>
</semantics>
</math></span><img src="./f066e981e530bacc07efc6a10fa82deee985929e.svg" 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 }" loading="lazy"></span>)-good if
</p>
<dl><dd>• <span class="mwe-math-element mwe-math-element-inline"><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="./132e57acb643253e7810ee9702d9581f159a1c61.svg" 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}" loading="lazy"></span> has <a href="Degree_of_a_polynomial" title="Degree of a polynomial">degree</a> <span class="mwe-math-element mwe-math-element-inline"><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="./7bf1b7e1e8ed2801660c299494ed10e6f18f790d.svg" 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}" loading="lazy"></span>,</dd></dl>
<dl><dd>• there exist distinct <a href="Subset" title="Subset">subsets</a> <span class="mwe-math-element mwe-math-element-inline"><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<!-- ℓ --></mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A_{1},\ldots ,A_{\ell }}</annotation>
</semantics>
</math></span><img src="./de49fb0a3e0438e6eea8208984b6488db99538c5.svg" 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 }}" loading="lazy"></span> of <span class="mwe-math-element mwe-math-element-inline"><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="./dbb96e056c071d13fc7702013f9273e7f5cd88a7.svg" 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}}" loading="lazy"></span> such that</dd></dl>
<dl><dd><dl><dd>– for any <span class="mwe-math-element mwe-math-element-inline"><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<!-- 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="./0944b6fbd3ab232ffeee17bed5a847132afc4575.svg" 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 \}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><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="./65d7e926fc46d61b39b462b66576064174da1e38.svg" 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}\}}" loading="lazy"></span> for some <span class="mwe-math-element mwe-math-element-inline"><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="./5c8760350012a306d2565e374be4a56fb7f42a87.svg" 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}}" loading="lazy"></span> , i.e., <span class="mwe-math-element mwe-math-element-inline"><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="./132e57acb643253e7810ee9702d9581f159a1c61.svg" 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}" loading="lazy"></span> is <a href="Constant_(mathematics)" title="Constant (mathematics)">constant</a> on <span class="mwe-math-element mwe-math-element-inline"><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="./1aed3b5def921afbe6cc48aaf8f9b11c6f1c1e2d.svg" 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}}" loading="lazy"></span>,</dd></dl></dd></dl>
<dl><dd><dl><dd>– <span class="mwe-math-element mwe-math-element-inline"><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="./a86acce714912dc60b59a24f617dc969868b01a5.svg" 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}" loading="lazy"></span>,</dd></dl></dd></dl>
<dl><dd><dl><dd>– <span class="mwe-math-element mwe-math-element-inline"><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="./a970bc33f4f140ab7fe3e0bee5435f86ec0cf4a6.svg" 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 }" loading="lazy"></span> for any <span class="mwe-math-element mwe-math-element-inline"><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="./d95aeb406bb427ac96806bc00c30c91d31b858be.svg" 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}" loading="lazy"></span>.</dd></dl></dd></dl>
<p>We say that {<span class="mwe-math-element mwe-math-element-inline"><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<!-- ℓ --></mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A_{1},\ldots ,A_{\ell }}</annotation>
</semantics>
</math></span><img src="./de49fb0a3e0438e6eea8208984b6488db99538c5.svg" 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 }}" loading="lazy"></span>} is a splitting covering for <span class="mwe-math-element mwe-math-element-inline"><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="./132e57acb643253e7810ee9702d9581f159a1c61.svg" 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}" loading="lazy"></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">Tamo–Barg construction</h3></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 mwe-math-element-inline"><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<!-- ℓ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (r,\ell )}</annotation>
</semantics>
</math></span><img src="./8638c0b218440f24dc34234ae9c28f20165db910.svg" 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 )}" loading="lazy"></span>-good polynomial <span class="mwe-math-element mwe-math-element-inline"><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="./202945cce41ecebb6f643f31d119c514bec7a074.svg" 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)}" loading="lazy"></span> over <span class="mwe-math-element mwe-math-element-inline"><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="./dbb96e056c071d13fc7702013f9273e7f5cd88a7.svg" 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}}" loading="lazy"></span> is given with splitting covering <span class="mwe-math-element mwe-math-element-inline"><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<!-- 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="./0944b6fbd3ab232ffeee17bed5a847132afc4575.svg" 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 \}}" loading="lazy"></span>.</dd>
<dd>• Let <span class="mwe-math-element mwe-math-element-inline"><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<!-- 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="./ba6ac878a12a37866518036599292a7fe20c31ab.svg" 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}" loading="lazy"></span> be a positive <a href="Integer" title="Integer">integer</a>.</dd>
<dd>• Consider the following <span class="mwe-math-element mwe-math-element-inline"><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="./dbb96e056c071d13fc7702013f9273e7f5cd88a7.svg" 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}}" loading="lazy"></span>-<a href="Vector_space" title="Vector space">vector space</a> of <a href="Polynomial" title="Polynomial">polynomials</a> <span class="mwe-math-element mwe-math-element-block"><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=\left\{\sum _{i=0}^{s}g_{i}(x)f(x)^{i}:\deg(g_{i}(x))\leq \deg(f(x))-2\right\}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
<mo>=</mo>
<mrow>
<mo>{</mo>
<mrow>
<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>
</mrow>
<mo>}</mo>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V=\left\{\sum _{i=0}^{s}g_{i}(x)f(x)^{i}:\deg(g_{i}(x))\leq \deg(f(x))-2\right\}.}</annotation>
</semantics>
</math></span></span></dd>
<dd>• Let <span class="mwe-math-element mwe-math-element-inline"><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<!-- 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="./f3285581251451513635aad6aa87fb2b92b89116.svg" 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}}" loading="lazy"></span>.</dd>
<dd>• The code <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\operatorname {ev} _{T}(g):g\in V\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>ev</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msub>
<mo><!-- --></mo>
<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 \{\operatorname {ev} _{T}(g):g\in V\}}</annotation>
</semantics>
</math></span><img src="./bbf5e889047570bc391031fa2c1b89e08c5a11f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.58ex; height:2.843ex;" alt="{\displaystyle \{\operatorname {ev} _{T}(g):g\in V\}}" loading="lazy"></span> is an <span class="mwe-math-element mwe-math-element-inline"><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<!-- 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="./72339861e98c4ae93d1cbd12fee3c4276eae2b62.svg" 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)}" loading="lazy"></span>-optimal locally coverable code, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {ev} _{T}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ev</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {ev} _{T}}</annotation>
</semantics>
</math></span><img src="./65ea2084424b110a5d2e2cbf2301dff01764a00a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.649ex; height:2.009ex;" alt="{\displaystyle \operatorname {ev} _{T}}" loading="lazy"></span> denotes evaluation of <span class="mwe-math-element mwe-math-element-inline"><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="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" 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}" loading="lazy"></span> at all points in the <a href="Set_(mathematics)" title="Set (mathematics)">set</a> <span class="mwe-math-element mwe-math-element-inline"><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="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" 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}" loading="lazy"></span>.</dd></dl>
<div class="mw-heading mw-heading3"><h3 id="Parameters_of_Tamo–Barg_codes">Parameters of Tamo–Barg codes</h3></div>
<dl><dd>• <b>Length.</b> The length is the number of evaluation points. Because the <a href="Set_(mathematics)" title="Set (mathematics)">sets</a> <span class="mwe-math-element mwe-math-element-inline"><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="./1aed3b5def921afbe6cc48aaf8f9b11c6f1c1e2d.svg" 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}}" loading="lazy"></span> are <a href="Disjoint_sets" title="Disjoint sets">disjoint</a> for <span class="mwe-math-element mwe-math-element-inline"><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<!-- 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="./0944b6fbd3ab232ffeee17bed5a847132afc4575.svg" 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 \}}" loading="lazy"></span>, the length of the code is <span class="mwe-math-element mwe-math-element-inline"><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<!-- ℓ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |T|=(r+1)\ell }</annotation>
</semantics>
</math></span><img src="./eeb2d4817facb01e702506c01baa4c87cda5a208.svg" 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 }" loading="lazy"></span>.</dd></dl>
<dl><dd>• <b>Dimension.</b> The <a href="Dimension" title="Dimension">dimension</a> of the code is <span class="mwe-math-element mwe-math-element-inline"><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="./5ee48f5cbb9020242def022e704997572e5a8bae.svg" 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}" loading="lazy"></span>, for <span class="mwe-math-element mwe-math-element-inline"><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="./01d131dfd7673938b947072a13a9744fe997e632.svg" 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}" loading="lazy"></span> ≤ <span class="mwe-math-element mwe-math-element-inline"><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<!-- ℓ --></mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \ell -1}</annotation>
</semantics>
</math></span><img src="./8144f37f93f6f504a155fa04a07f0da8917fab1d.svg" 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}" loading="lazy"></span>, as each <span class="mwe-math-element mwe-math-element-inline"><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="./2ce36142a0a1c6660e82bdf3ef3f1551317efe0c.svg" 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}}" loading="lazy"></span> has <a href="Degree_of_a_polynomial" title="Degree of a polynomial">degree</a> at most <span class="mwe-math-element mwe-math-element-inline"><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="./d9adf41a4343bc036dd08ef28cad637148f98f8a.svg" 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}" loading="lazy"></span>, covering a <a href="Vector_space" title="Vector space">vector space</a> of <a href="Dimension" title="Dimension">dimension</a> <span class="mwe-math-element mwe-math-element-inline"><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="./33caf14249cfa37b1d06215ac2d5587c0dbd1e10.svg" 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}" loading="lazy"></span>, and by the construction of <span class="mwe-math-element mwe-math-element-inline"><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="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" 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}" loading="lazy"></span>, there are <span class="mwe-math-element mwe-math-element-inline"><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="./cc3e2e7deb52c40fc144183ba49b8494015640ec.svg" 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}" loading="lazy"></span> distinct <span class="mwe-math-element mwe-math-element-inline"><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="./2ce36142a0a1c6660e82bdf3ef3f1551317efe0c.svg" 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}}" loading="lazy"></span>.</dd></dl>
<dl><dd>• <b>Distance.</b> The <a href="Hamming_distance" title="Hamming distance">distance</a> is given by the fact that <span class="mwe-math-element mwe-math-element-inline"><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="./e41752968e97415360d5636f1ae7ab3e7ea6c373.svg" 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}}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><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="./a9a90aee875b487766cc0fcfef677788bc36a2a6.svg" 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)}" loading="lazy"></span>, and the obtained code is the <a href="Reed%E2%80%93Solomon_error_correction" title="Reed–Solomon error correction">Reed-Solomon code</a> of <a href="Degree_of_a_polynomial" title="Degree of a polynomial">degree</a> at most <span class="mwe-math-element mwe-math-element-inline"><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="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" 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}" loading="lazy"></span>, so the minimum <a href="Hamming_distance" title="Hamming distance">distance</a> equals <span class="mwe-math-element mwe-math-element-inline"><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<!-- 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="./8ccc4b43573af51eea913ef097b89f338c89af49.svg" 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))}" loading="lazy"></span>.</dd></dl>
<dl><dd>• <b>Locality.</b> After the erasure of the single component, the evaluation at <span class="mwe-math-element mwe-math-element-inline"><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="./9498f2f918d0b59e82c9346aa62925f06ff9fae8.svg" 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}}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><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="./aa3b1d6885c3f48010e69a1bee91ea2ce779f52c.svg" 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}" loading="lazy"></span>, is unknown, but the evaluations for all other <span class="mwe-math-element mwe-math-element-inline"><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="./d0a291c98d6222aa149241b038ef255ca609a6f3.svg" 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}}" loading="lazy"></span> are known, so at most <span class="mwe-math-element mwe-math-element-inline"><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="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" 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}" loading="lazy"></span> evaluations are needed to uniquely determine the erased component, which gives us the locality of <span class="mwe-math-element mwe-math-element-inline"><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="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" 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}" loading="lazy"></span>.</dd></dl>
<dl><dd>To see this, <span class="mwe-math-element mwe-math-element-inline"><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="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" 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}" loading="lazy"></span> restricted to <span class="mwe-math-element mwe-math-element-inline"><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="./6019bb70c912e59e9d5f442e9217517743ed4831.svg" 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}}" loading="lazy"></span> can be described by a <a href="Polynomial" title="Polynomial">polynomial</a> <span class="mwe-math-element mwe-math-element-inline"><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="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" 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}" loading="lazy"></span> of <a href="Degree_of_a_polynomial" title="Degree of a polynomial">degree</a> at most <span class="mwe-math-element mwe-math-element-inline"><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="./2ac1494980148eb5c55cdf58c8da0a7dab5d1413.svg" 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}" loading="lazy"></span> thanks to the form of the <a href="Element_(mathematics)" title="Element (mathematics)">elements</a> in <span class="mwe-math-element mwe-math-element-inline"><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="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" 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}" loading="lazy"></span> (i.e., thanks to the fact that <span class="mwe-math-element mwe-math-element-inline"><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="./132e57acb643253e7810ee9702d9581f159a1c61.svg" 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}" loading="lazy"></span> is <a href="Constant_function" title="Constant function">constant</a> on <span class="mwe-math-element mwe-math-element-inline"><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="./6019bb70c912e59e9d5f442e9217517743ed4831.svg" 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}}" loading="lazy"></span>, and the <span class="mwe-math-element mwe-math-element-inline"><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="./2ce36142a0a1c6660e82bdf3ef3f1551317efe0c.svg" 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}}" loading="lazy"></span>'s have <a href="Degree_of_a_polynomial" title="Degree of a polynomial">degree</a> at most <span class="mwe-math-element mwe-math-element-inline"><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="./d9adf41a4343bc036dd08ef28cad637148f98f8a.svg" 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}" loading="lazy"></span>). On the other hand <span class="mwe-math-element mwe-math-element-inline"><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="./2bccb9b29359f46c286621f318a48a282a33bda1.svg" 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}" loading="lazy"></span>, and <span class="mwe-math-element mwe-math-element-inline"><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="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" 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}" loading="lazy"></span> evaluations uniquely determine a <a href="Polynomial" title="Polynomial">polynomial</a> of <a href="Degree_of_a_polynomial" title="Degree of a polynomial">degree</a> <span class="mwe-math-element mwe-math-element-inline"><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="./1ad04896104e929da32fad148e240b3fd8dfa874.svg" 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}" loading="lazy"></span>. Therefore <span class="mwe-math-element mwe-math-element-inline"><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="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" 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}" loading="lazy"></span> can be constructed and evaluated at <span class="mwe-math-element mwe-math-element-inline"><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="./d0096fb78d6843c9fb67a840dc796b61ad93eec2.svg" 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}}" loading="lazy"></span> to recover <span class="mwe-math-element mwe-math-element-inline"><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="./62364e19c48261fc33af6fab95fed3c053446b8c.svg" 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})}" loading="lazy"></span>.</dd></dl>
<div class="mw-heading mw-heading3"><h3 id="Example_of_Tamo–Barg_construction">Example of Tamo–Barg construction</h3></div>
<p>We will use <span class="mwe-math-element mwe-math-element-inline"><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="./66b40ce98ea0fe92cf62a531dc6586f1437c7873.svg" 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]}" loading="lazy"></span> to construct <span class="mwe-math-element mwe-math-element-inline"><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="./8153c186e2aa3ba4f0f7836daf3f1580d3710d87.svg" 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]}" loading="lazy"></span>-LRC. Notice that the <a href="Degree_of_a_polynomial" title="Degree of a polynomial">degree</a> of this <a href="Polynomial" title="Polynomial">polynomial</a> is 5, and it is <a href="Constant_(mathematics)" title="Constant (mathematics)">constant</a> on <span class="mwe-math-element mwe-math-element-inline"><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="./1aed3b5def921afbe6cc48aaf8f9b11c6f1c1e2d.svg" 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}}" loading="lazy"></span> for <span class="mwe-math-element mwe-math-element-inline"><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="./7d14e2eeadd41cab498745e32e529fd68459cfb3.svg" 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\}}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><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="./c0b3b1a9eccc40489f0558a78521957bad258c39.svg" 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\}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><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="./7bb37be079e2ba051a638c8bf7f9d1f2d96b5de0.svg" 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}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><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="./6dbef0e6646ae45afbadf9644254c37e5f4fb7af.svg" 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}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><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="./115ea1bb6f4e62c51753f486d1c02119a92c36b8.svg" 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}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><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="./2f758cafdd2f469413e7bae4728b69639110157e.svg" 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}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><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="./3085500910cc915b4f4c11d29e32ffb564e74801.svg" 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}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><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="./0a76aeb1048321fa13b37f68918d3667b251dd1e.svg" 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}}" loading="lazy"></span>, and <span class="mwe-math-element mwe-math-element-inline"><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="./bde4776ef4afd6b784855601b812aa5dc8f3b31e.svg" 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}}" loading="lazy"></span>: <span class="mwe-math-element mwe-math-element-inline"><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="./f9b85bf0321cfe79a5d3b75b81ce9a72b3962a36.svg" 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\}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><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="./b2585a742016f3ba7f547193c753ec961559f405.svg" 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\}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><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="./defdd46d166245cce180fb2e3ade17e637ea93e3.svg" 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\}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><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="./ed18ee1a12b75f0d7f66d8162712d898773e0a92.svg" 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\}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><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="./b79d45b796de45a350ac5454fb7e881f2b6f2a43.svg" 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\}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><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="./175de8423dd111977467ed4dae5809b7c78745b9.svg" 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\}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><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="./c9f7764dea2aa55478ce40712f80398e78714656.svg" 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\}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><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="./dd9d339ff522547fe5d6f167755eaa82d96ed922.svg" 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\}}" loading="lazy"></span>. Hence, <span class="mwe-math-element mwe-math-element-inline"><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="./9e07d6b7e67f8ecc28b9c3206f192b4144168b61.svg" 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}}" loading="lazy"></span> is a <span class="mwe-math-element mwe-math-element-inline"><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="./c34b45045069c6301c7aec705b6392d3eb37aea4.svg" 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)}" loading="lazy"></span>-good polynomial over <span class="mwe-math-element mwe-math-element-inline"><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="./a6107bfc7b73109409c793ab5d8caa99271af811.svg" 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}}" loading="lazy"></span> by the definition. Now, we will use this <a href="Polynomial" title="Polynomial">polynomial</a> to construct a code of <a href="Dimension" title="Dimension">dimension</a> <span class="mwe-math-element mwe-math-element-inline"><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="./1170deafc5d96c9d76fcd097806d334487cddc1f.svg" 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}" loading="lazy"></span> and length <span class="mwe-math-element mwe-math-element-inline"><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="./15376579267b277377fc09465f44e0ec1c761891.svg" 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}" loading="lazy"></span> over <span class="mwe-math-element mwe-math-element-inline"><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="./a6107bfc7b73109409c793ab5d8caa99271af811.svg" 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}}" loading="lazy"></span>. The locality of this code is 4, which will allow us to recover a single <a href="Server_(computing)" title="Server (computing)">server</a> failure by looking at the information contained in at most 4 other <a href="Server_(computing)" title="Server (computing)">servers</a>.
</p><p>Next, let us define the encoding <a href="Polynomial" title="Polynomial">polynomial</a>: <span class="mwe-math-element mwe-math-element-inline"><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="./0138d01ee8ba8fb1c3b67d54e80c95d9421226c8.svg" 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}}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><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="./326ab12594b5e4b8afa24311b60e1fb503553510.svg" 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}}" loading="lazy"></span>. So, <span class="mwe-math-element mwe-math-element-inline"><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="./46d07ad34a6d88297f93b7c5a4e3820c9d68b2e9.svg" 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)=}" loading="lazy"></span> <span class="mwe-math-element mwe-math-element-inline"><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="./46971d01639d66fbbcac9fc54a568d525513c823.svg" 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}+}" loading="lazy"></span> <span class="mwe-math-element mwe-math-element-inline"><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="./2e8ab77bad50c43aaa51d58edb53930ad703c1cd.svg" 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}+}" loading="lazy"></span> <span class="mwe-math-element mwe-math-element-inline"><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="./17622918ed995192ad52b357ed39c0ef0845eefb.svg" 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+}" loading="lazy"></span> <span class="mwe-math-element mwe-math-element-inline"><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="./8be7b0fde0c5e30f3930d94351769137ae35c20b.svg" 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}+}" loading="lazy"></span> <span class="mwe-math-element mwe-math-element-inline"><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="./ef5db8c69b339c237dde5d6843b0a0a4242ee08b.svg" 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}+}" loading="lazy"></span> <span class="mwe-math-element mwe-math-element-inline"><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="./ee7c4dab1f3e0df69dd592a4f7b0c75701669481.svg" 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}+}" loading="lazy"></span> <span class="mwe-math-element mwe-math-element-inline"><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="./4400b47dc191ec0efc78a5203cd5fbace8c01737.svg" 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}+}" loading="lazy"></span> <span class="mwe-math-element mwe-math-element-inline"><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="./8d8e020028acd2f9d093e303a41f4ddb819755fa.svg" 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}}" loading="lazy"></span>.
</p><p>Thus, we can use the obtained encoding <a href="Polynomial" title="Polynomial">polynomial</a> if we take our <a href="Data" title="Data">data</a> to encode as the <a href="Row_and_column_vectors" title="Row and column vectors">row vector</a> <span class="mwe-math-element mwe-math-element-inline"><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="./8471c90b288179a1f6eb4fc06d65bad0b1f43f8d.svg" 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=}" loading="lazy"></span> <span class="mwe-math-element mwe-math-element-inline"><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="./d297f927661cd7c79a9b7c5362ac4b613b156b1b.svg" 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})}" loading="lazy"></span>. Encoding the <a href="Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vector</a> <span class="mwe-math-element mwe-math-element-inline"><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="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" 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}" loading="lazy"></span> to a length 15 message <a href="Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vector</a> <span class="mwe-math-element mwe-math-element-inline"><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="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" 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}" loading="lazy"></span> by multiplying <span class="mwe-math-element mwe-math-element-inline"><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="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" 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}" loading="lazy"></span> by the <a href="Generator_matrix" title="Generator matrix">generator matrix</a>
</p>
<dl><dd><div style="text-align: center;"><span class="mwe-math-element mwe-math-element-inline"><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="./afd1be30d2831bbe7dc465323198e803c05ff3ec.svg" 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}}.}" loading="lazy"></span></div></dd></dl>
<p>For example, the encoding of information <a href="Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vector</a> <span class="mwe-math-element mwe-math-element-inline"><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="./47a3f040f2765026dcdb8aa52f793f428aaa260d.svg" 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)}" loading="lazy"></span> gives the codeword <span class="mwe-math-element mwe-math-element-inline"><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="./418183828390aeebd328057e6cccf82b44cebc25.svg" 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)}" loading="lazy"></span>.
</p><p>Observe that we constructed an optimal LRC; therefore, using the <a href="Singleton_bound" title="Singleton bound">Singleton bound</a>, we have that the <a href="Hamming_distance" title="Hamming distance">distance</a> of this code is <span class="mwe-math-element mwe-math-element-inline"><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="./deab1ccefd19762d54bf7c8afbf635d4957d1607.svg" 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}" loading="lazy"></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></div>
<p>A code <span class="mwe-math-element mwe-math-element-inline"><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="./4fc55753007cd3c18576f7933f6f089196732029.svg" 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}" loading="lazy"></span> has all-symbol locality <span class="mwe-math-element mwe-math-element-inline"><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="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" 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}" loading="lazy"></span> and availability <span class="mwe-math-element mwe-math-element-inline"><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="./65658b7b223af9e1acc877d848888ecdb4466560.svg" 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}" loading="lazy"></span> if every code symbol can be recovered from <span class="mwe-math-element mwe-math-element-inline"><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="./65658b7b223af9e1acc877d848888ecdb4466560.svg" 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}" loading="lazy"></span> disjoint repair sets of other symbols, each <a href="Set_(mathematics)" title="Set (mathematics)">set</a> of <a href="Cardinality" title="Cardinality">size</a> at most <span class="mwe-math-element mwe-math-element-inline"><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="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" 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}" loading="lazy"></span> symbols. Such codes are called <span class="mwe-math-element mwe-math-element-inline"><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="./5c80cb006f630d1af28daf2db5dd83fd47fba76c.svg" 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}}" loading="lazy"></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="Hamming_distance" title="Hamming distance">distance</a> of <span class="mwe-math-element mwe-math-element-inline"><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="./5eef76e1d6b74cc535236a6249e75afeea540ee3.svg" 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}}" loading="lazy"></span>-LRC having locality <span class="mwe-math-element mwe-math-element-inline"><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="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" 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}" loading="lazy"></span> and availability <span class="mwe-math-element mwe-math-element-inline"><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="./65658b7b223af9e1acc877d848888ecdb4466560.svg" 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}" loading="lazy"></span> satisfies the <a href="Upper_and_lower_bounds" title="Upper and lower bounds">upper bound</a>
</p>
<div style="text-align: center;"><span class="mwe-math-element mwe-math-element-inline"><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-\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="./61e057434b351a5356b936564b12b333f2c354a3.svg" class="mwe-math-fallback-image-inline 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 .}" loading="lazy"></span></div>
<p>If the code is <a href="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 mwe-math-element-inline"><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="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" 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}" loading="lazy"></span> and availability <span class="mwe-math-element mwe-math-element-inline"><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="./65658b7b223af9e1acc877d848888ecdb4466560.svg" 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}" loading="lazy"></span>, and is called <span class="mwe-math-element mwe-math-element-inline"><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="./47d303df46de33e2cda0dd8d43f6d4ca8e7d58f0.svg" 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}}" loading="lazy"></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="Hamming_distance" title="Hamming distance">distance</a> <span class="mwe-math-element mwe-math-element-inline"><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="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" 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}" loading="lazy"></span> of an <span class="mwe-math-element mwe-math-element-inline"><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="./5eef76e1d6b74cc535236a6249e75afeea540ee3.svg" 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}}" loading="lazy"></span> linear <span class="mwe-math-element mwe-math-element-inline"><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="./47d303df46de33e2cda0dd8d43f6d4ca8e7d58f0.svg" 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}}" loading="lazy"></span>-LRC satisfies the <a href="Upper_and_lower_bounds" title="Upper and lower bounds">upper bound</a>
</p>
<div style="text-align: center;"><span class="mwe-math-element mwe-math-element-inline"><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 {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="./9b3404e8c94be7f889a7c2706db9748d71701fe9.svg" class="mwe-math-fallback-image-inline 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.}" loading="lazy"></span></div>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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</style><cite id="CITEREFPapailiopoulosDimakis2012" class="citation cs2">Papailiopoulos, Dimitris S.; Dimakis, Alexandros G. (2012), "Locally repairable codes", <i>2012 IEEE International Symposium on Information Theory Proceedings</i>, Cambridge, MA, USA: IEEE, pp. <span class="nowrap">2771–</span>2775, <a href="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="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="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4673-2579-0</bdi></cite></span>
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<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFHaymakerMalmskogMatthews2022" class="citation cs2">Haymaker, K.; Malmskog, B.; Matthews, G. (2022), <i>Locally Recoverable Codes with Availability </i>t<i>≥2 from Fiber Products of Curves</i>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.3934%2Famc.2018020">10.3934/amc.2018020</a></span></cite></span>
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<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFPapailiopoulosDimakis2012" class="citation cs2">Papailiopoulos, Dimitris S.; Dimakis, Alexandros G. (2012), "Locally repairable codes", <i>2012 IEEE International Symposium on Information Theory</i>, Cambridge, MA, USA, pp. <span class="nowrap">2771–</span>2775, <a href="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="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="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4673-2579-0</bdi></cite><span class="cs1-maint citation-comment"><code class="cs1-code">{{citation}}</code>: CS1 maint: location missing publisher (link)</span></span>
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