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Covering code

Set of "near" codewords in coding theory

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Source updated July 5, 2026

In coding theory, 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.

01Definition

Let $q\geq 2$ , $n\geq 1$ , $R\geq 0$ be integers. A code $C\subseteq Q^{n}$ over an alphabet Q of size |Q| = q is called q-ary R-covering code of length n if for every word $y\in Q^{n}$ there is a codeword $x\in C$ such that the Hamming distance $d_{H}(x,y)\leq R$ . In other words, the spheres (or balls or rook-domains) of radius R with respect to the Hamming metric around the codewords of C have to exhaust the finite metric space $Q^{n}$ . The covering radius of a code C is the smallest R such that C is R-covering. Every perfect code is a covering code of minimal size.

02Example

C = {0134,0223,1402,1431,1444,2123,2234,3002,3310,4010,4341} is a 5-ary 2-covering code of length 4.

03Covering problem

The determination of the minimal size $K_{q}(n,R)$ of a q-ary R-covering code of length n is a very hard problem. In many cases, only upper and lower bounds are known with a large gap between them. Every construction of a covering code gives an upper bound on K_q(n, R). Lower bounds include the sphere covering bound and Rodemich's bounds $K_{q}(n,1)\geq q^{n-1}/(n-1)$ and $K_{q}(n,n-2)\geq q^{2}/(n-1)$ . The covering problem is closely related to the packing problem in $Q^{n}$ , i.e. the determination of the maximal size of a q-ary e-error correcting code of length n.

04Football pools problem

A particular case is the football pools problem, based on football pool 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 $n={\tfrac {1}{2}}(3^{k}-1)$ then 3^n-k are needed, so for n = 4, k = 2, 9 are needed; for n = 13, k = 3, 59049 are needed. The best bounds known as of 2011 are

n	1	2	3	4	5	6	7	8	9	10	11	12	13	14
K₃(n,1)	1	3	5	9	27	71-73	156-186	402-486	1060-1269	2854-3645	7832-9477	21531-27702	59049	166610-177147
K₃(n,2)		1	3	3	8	15-17	26-34	54-81	130-219	323-555	729	1919-2187	5062-6561	12204-19683
K₃(n,3)			1	3	3	6	11-12	14-27	27-54	57-105	117-243	282-657	612-1215	1553-2187

05Applications

The standard work on covering codes lists the following applications.

Compression with distortion
Data compression
Decoding errors and erasures
Broadcasting in interconnection networks
Football pools
Write-once memories
Berlekamp-Gale game
Speech coding
Cellular telecommunications
Subset sums and Cayley graphs

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Sources and credits

This article is adapted from the Wikipedia article “Covering code”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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