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<title>Covering code</title>

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Covering code

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">In <a href="Coding_theory" title="Coding theory">coding theory</a>, a covering code is a set of elements (called codewords) in a space, with the property that every element of the space is within a fixed distance of some codeword.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\geq 2}">

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<annotation encoding="application/x-tex">{\displaystyle q\geq 2}</annotation>

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</math><img src="./d65b649e4e4632de8002b4b7d1dee1675207ffe0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.33ex; height:2.509ex;" alt="{\displaystyle q\geq 2}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 1}">

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<annotation encoding="application/x-tex">{\displaystyle n\geq 1}</annotation>

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</math><img src="./d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\geq 0}">

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<annotation encoding="application/x-tex">{\displaystyle R\geq 0}</annotation>

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</math><img src="./56c27fc5e613e042a8189e48f4b5edec9acda98b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.025ex; height:2.343ex;" alt="{\displaystyle R\geq 0}" loading="lazy"> be <a href="Integers" class="mw-redirect" title="Integers">integers</a>.

A <a href="Code" title="Code">code</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C\subseteq Q^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle C\subseteq Q^{n}}</annotation>

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</math><img src="./6b52f76266f592813eb0822da20309e3bbdae00d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.922ex; height:2.676ex;" alt="{\displaystyle C\subseteq Q^{n}}" loading="lazy"> over an <a href="Alphabet" title="Alphabet">alphabet</a> Q of size |Q| = q is called

q-ary R-covering code of length n

if for every word <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in Q^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle y\in Q^{n}}</annotation>

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</math><img src="./60a2004566924a27203628337081a05be966161f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.053ex; height:2.676ex;" alt="{\displaystyle y\in Q^{n}}" loading="lazy"> there is a <a href="Code_word_(communication)" title="Code word (communication)">codeword</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in C}">

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<annotation encoding="application/x-tex">{\displaystyle x\in C}</annotation>

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</math><img src="./f7fc788379bff7289bdf694ffd68ee690e999eb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.937ex; height:2.176ex;" alt="{\displaystyle x\in C}" loading="lazy">

such that the <a href="Hamming_distance" title="Hamming distance">Hamming distance</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{H}(x,y)\leq R}">

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<annotation encoding="application/x-tex">{\displaystyle d_{H}(x,y)\leq R}</annotation>

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</math><img src="./1193d3267713f812fee51b534c5dad40f5d6fab5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.091ex; height:2.843ex;" alt="{\displaystyle d_{H}(x,y)\leq R}" loading="lazy">.

In other words, the <a href="Spheres" class="mw-redirect" title="Spheres">spheres</a> (or <a href="Ball_(mathematics)" title="Ball (mathematics)">balls</a> or rook-domains) of <a href="Radius" title="Radius">radius</a> R

with respect to the Hamming <a href="Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a> around the codewords of C have to exhaust

the <a href="https://en.wiktionary.org/wiki/finite" class="extiw external" title="wikt:finite">finite</a> <a href="Metric_space" title="Metric space">metric space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle Q^{n}}</annotation>

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The <a href="Covering_radius" class="mw-redirect" title="Covering radius">covering radius</a> of a code C is the smallest R such that C is R-covering.

Every <a href="Perfect_code" class="mw-redirect" title="Perfect code">perfect code</a> is a covering code of minimal size.

<div class="mw-heading mw-heading2"><h2 id="Example">Example</h2></div>

C = {0134,0223,1402,1431,1444,2123,2234,3002,3310,4010,4341} is a 5-ary 2-covering code of length 4.<a href="#cite_note-1">[1]</a>

<div class="mw-heading mw-heading2"><h2 id="Covering_problem">Covering problem</h2></div>

The determination of the minimal size <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{q}(n,R)}">

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<annotation encoding="application/x-tex">{\displaystyle K_{q}(n,R)}</annotation>

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</math><img src="./0fa9ea7a67c91c65e7d9ba584d2d8215cf5602a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.964ex; height:3.009ex;" alt="{\displaystyle K_{q}(n,R)}" loading="lazy"> of a q-ary R-covering code of length n is a very hard problem. In many cases, only <a href="Upper_and_lower_bounds" title="Upper and lower bounds">upper and lower bounds</a> are known with a large gap between them.

Every construction of a covering code gives an upper bound on Kq(n, R).

Lower bounds include the sphere covering bound and

Rodemich's bounds <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{q}(n,1)\geq q^{n-1}/(n-1)}">

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<annotation encoding="application/x-tex">{\displaystyle K_{q}(n,1)\geq q^{n-1}/(n-1)}</annotation>

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</math><img src="./c640cabf6fd4476030400ebc747b34ee9f5690c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.228ex; height:3.343ex;" alt="{\displaystyle K_{q}(n,1)\geq q^{n-1}/(n-1)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{q}(n,n-2)\geq q^{2}/(n-1)}">

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<annotation encoding="application/x-tex">{\displaystyle K_{q}(n,n-2)\geq q^{2}/(n-1)}</annotation>

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</math><img src="./92cdcb7ebb4fb7ea0ae29cdcab0f73428b4dc033.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.199ex; height:3.343ex;" alt="{\displaystyle K_{q}(n,n-2)\geq q^{2}/(n-1)}" loading="lazy">.<a href="#cite_note-2">[2]</a>

The covering problem is closely related to the packing problem in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q^{n}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle Q^{n}}</annotation>

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</math><img src="./5da2f2952f8145669909f96e7f344e533b1e9a92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.057ex; height:2.676ex;" alt="{\displaystyle Q^{n}}" loading="lazy">, i.e. the determination of the maximal size of a q-ary e-<a href="Error_detection_and_correction" title="Error detection and correction">error correcting</a> code of length n.

<div class="mw-heading mw-heading2"><h2 id="Football_pools_problem">Football pools problem</h2></div>

A particular case is the football pools problem, based on <a href="Football_pool" class="mw-redirect" title="Football pool">football pool</a> betting, where the aim is to come up with a betting system over n football matches that, regardless of the outcome, has at most R 'misses'. Thus, for n matches with at most one 'miss', a ternary covering, K3(n,1), is sought.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n={\tfrac {1}{2}}(3^{k}-1)}">

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle n={\tfrac {1}{2}}(3^{k}-1)}</annotation>

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</math><img src="./6f20c54891c917f111b11f4f9b93dec12615cf2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.215ex; height:3.509ex;" alt="{\displaystyle n={\tfrac {1}{2}}(3^{k}-1)}" loading="lazy"> then 3n-k are needed, so for n = 4, k = 2, 9 are needed; for n = 13, k = 3, 59049 are needed.<a href="#cite_note-3">[3]</a> The best bounds known as of 2011<a href="#cite_note-4">[4]</a> are

<th>n

</th>

<th>1

</th>

<th>2

</th>

<th>3

</th>

<th>4

</th>

<th>5

</th>

<th>6

</th>

<th>7

</th>

<th>8

</th>

<th>9

</th>

<th>10

</th>

<th>11

</th>

<th>12

</th>

<th>13

</th>

<th>14

</th></tr>

<tr>

<th>K3(n,1)

</th>

<td>1

</td>

<td>3

</td>

<td>5

</td>

<td>9

</td>

<td>27

</td>

<td>71-73

</td>

<td>156-186

</td>

<td>402-486

</td>

<td>1060-1269

</td>

<td>2854-3645

</td>

<td>7832-9477

</td>

<td>21531-27702

</td>

<td>59049

</td>

<td>166610-177147

</td></tr>

<tr>

<th>K3(n,2)

</th>

<td>

</td>

<td>1

</td>

<td>3

</td>

<td>3

</td>

<td>8

</td>

<td>15-17

</td>

<td>26-34

</td>

<td>54-81

</td>

<td>130-219

</td>

<td>323-555

</td>

</td>

<td>1919-2187

</td>

<td>5062-6561

</td>

<td>12204-19683

</td></tr>

<tr>

<th>K3(n,3)

</th>

<td>

</td>

<td>

</td>

<td>1

</td>

<td>3

</td>

<td>3

</td>

<td>6

</td>

<td>11-12

</td>

<td>14-27

</td>

<td>27-54

</td>

<td>57-105

</td>

<td>117-243

</td>

<td>282-657

</td>

<td>612-1215

</td>

<td>1553-2187

</td></tr></tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

The standard work<a href="#cite_note-5">[5]</a> on covering codes lists the following applications.

<ul><li>Compression with <a href="Distortion" title="Distortion">distortion</a></li>

<li><a href="Data_compression" title="Data compression">Data compression</a></li>

<li><a href="Code" title="Code">Decoding</a> errors and erasures</li>

<li><a href="Broadcasting" title="Broadcasting">Broadcasting</a> in interconnection networks</li>

<li><a href="Football_pools" title="Football pools">Football pools</a><a href="#cite_note-6">[6]</a></li>

<li>Write-once memories</li>

<li>Berlekamp-Gale game</li>

<li><a href="Speech_coding" title="Speech coding">Speech coding</a></li>

<li>Cellular <a href="Telecommunications" title="Telecommunications">telecommunications</a></li>

<li><a href="Subset" title="Subset">Subset</a> sums and <a href="Cayley_graph" title="Cayley graph">Cayley graphs</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFP.R.J._Östergård1991" class="citation journal cs1">P.R.J. Östergård (1991). "Upper bounds for q-ary covering codes". <a href="IEEE_Transactions_on_Information_Theory" title="IEEE Transactions on Information Theory">IEEE Transactions on Information Theory</a>. 37: 660–664.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFE.R._Rodemich1970" class="citation journal cs1">E.R. Rodemich (1970). "Covering by rook-domains". <a href="Journal_of_Combinatorial_Theory" title="Journal of Combinatorial Theory">Journal of Combinatorial Theory</a>. 9: 117–128.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFKampsvan_Lint1967" class="citation journal cs1">Kamps, H.J.L.; van Lint, J.H. (December 1967). <a rel="nofollow" class="external text" href="http://alexandria.tue.nl/repository/freearticles/593454.pdf">"The football pool problem for 5 matches"</a> (PDF). Journal of Combinatorial Theory. 3 (4): 315–325. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0021-9800%2867%2980102-9">10.1016/S0021-9800(67)80102-9</a>. Retrieved 9 November 2022.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://old.sztaki.hu/~keri/codes/3_tables.pdf">"Bounds on K3(n, R) (lower and upper bounds on the size of ternary optimal covering codes)"</a> (PDF). SZÁMÍTÁSTECHNIKAI ÉS AUTOMATIZÁLÁSI KUTATÓINTÉZET. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20221027203847/http://old.sztaki.hu/~keri/codes/3_tables.pdf">Archived</a> (PDF) from the original on 27 October 2022. Retrieved 9 November 2022.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFG._Cohen,_I._Honkala,_S._Litsyn,_A._Lobstein1997" class="citation book cs1">G. Cohen, I. Honkala, S. Litsyn, A. Lobstein (1997). Covering Codes. <a href="Elsevier" title="Elsevier">Elsevier</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-444-82511-8</bdi>.</cite><code class="cs1-code">{{cite book}}</code>: CS1 maint: multiple names: authors list (link)

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFH._Hämäläinen,_I._Honkala,_S._Litsyn,_P.R.J._Östergård1995" class="citation journal cs1">H. Hämäläinen, I. Honkala, S. Litsyn, P.R.J. Östergård (1995). "Football pools — a game for mathematicians". <a href="American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a>. 102: 579–588.</cite><code class="cs1-code">{{cite journal}}</code>: CS1 maint: multiple names: authors list (link)

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://www.infres.enst.fr/~lobstein/biblio.html">Literature on covering codes</a></li>

<li><a rel="nofollow" class="external text" href="http://www.sztaki.hu/~keri/codes/index.htm">Bounds on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{q}(n,R)}">

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<annotation encoding="application/x-tex">{\displaystyle K_{q}(n,R)}</annotation>

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