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.

Definition

be integers. A code over an alphabet Q of size |Q| = q is called q-ary R-covering code of length n such that the Hamming distance. 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. 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.

Covering problem

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 Kq(n,R., i.e. the determination of the maximal size of a q-ary e-error correcting code of length n.

Football 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.

then 3n-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

Applications

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

References

External links

• Literature on covering codes