Covering code

In coding theory, a covering code is an object satisfying a certain mathematical property: A code of length n over Q is an R-covering code if for every word of Qⁿ there is a codeword such that their Hamming distance is $\le R$.

Definition

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 Qⁿ. 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.

Example

C = {0134,0223,1402,1431,1444,2123,2234,3002,3310,4010,4341} is a 5-ary 2-covering code of length 4.^[1]

Covering 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 lower and upper 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 $K_q(n,n-2)\geq q^2/(n-1)$.^[2] The covering problem is closely related to the packing problem in Qⁿ, i.e. the determination of the maximal size of a q-ary e-error correcting code of length n.

Applications

The standard work^[3] on covering codes lists the following applications.

• Compression with distortion
• Data compression
• Decoding errors and erasures
• Broadcasting in interconnection networks
• Football pools^[4]
• Write-once memories
• Berlekamp-Gale game
• Speech coding
• Cellular telecommunications
• Subset sums and Cayley graphs

References

1. ^ P.R.J. Östergård, Upper bounds for q-ary covering codes, IEEE Transactions on Information Theory, 37 (1991), 660-664
2. ^ E.R. Rodemich, Covering by rook-domains, Journal of Combinatorial Theory, 9 (1970), 117-128
3. ^ G. Cohen, I. Honkala, S. Litsyn, A. Lobstein, Covering Codes, Elsevier (1997) ISBN 0-444-82511-8
4. ^ H. Hämäläinen, I. Honkala, S. Litsyn, P.R.J. Östergård, Football pools - a game for mathematicians, American Mathematical Monthly, 102 (1995), 579-588

External links

• Literature on covering codes
• Bounds on K_q(n,R)