Covering code


In coding theory, a covering code is a set of elements in a space, with the property that every element of the space is within a fixed distance of some codeword.

Definition

Let,, be integers.
A code over an alphabet Q of size |Q| = q is called
q-ary R-covering code of length n
if for every word there is a codeword
such that the Hamming distance.
In other words, the spheres 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 = is a 5-ary 2-covering code of length 4.

Covering problem

The determination of the minimal size 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.
Lower bounds include the sphere covering bound and
Rodemich's bounds and.
The covering problem is closely related to the packing problem in, 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, is sought.
If 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
n1234567891011121314
K313592771-73156-186402-4861060-12692854-36457832-947721531-2770259049166610-177147
K3133815-1726-3454-81130-219323-5557291919-21875062-656112204-19683
K3133611-1214-2727-5457-105117-243282-657612-12151553-2187

Applications

The standard work on covering codes lists the following applications.