Covering code

Let ${\displaystyle q\geq 2}$, ${\displaystyle n\geq 1}$, ${\displaystyle R\geq 0}$ be integers. A code ${\displaystyle C\subseteq Q^{n}}$ over an alphabet Q of size |Q| = q is called q-ary R-covering code of length n if for every word ${\displaystyle y\in Q^{n}}$ there is a codeword ${\displaystyle x\in C}$ such that the Hamming distance ${\displaystyle d_{H}(x,y)\leq R}$. 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 ${\displaystyle Q^{n}}$. 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.

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

The determination of the minimal size ${\displaystyle K_{q}(n,R)}$ 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 ${\displaystyle K_{q}(n,1)\geq q^{n-1}/(n-1)}$ and ${\displaystyle K_{q}(n,n-2)\geq q^{2}/(n-1)}$.^[2] The covering problem is closely related to the packing problem in ${\displaystyle Q^{n}}$, i.e. the determination of the maximal size of a q-ary e-error correcting code of length n.

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 ${\displaystyle n={\tfrac {1}{2}}(3^{k}-1)}$ then 3^n-k are needed, so for n = 4, k = 2, 9 are needed; for n = 13, k = 3, 59049 are needed.^[3] The best bounds known as of 2011^[4] are

The standard work^[5] on covering codes lists the following applications.

• Compression with distortion
• Data compression
• Decoding errors and erasures
• Broadcasting in interconnection networks
• Football pools^[6]
• Write-once memories
• Berlekamp-Gale game
• Speech coding
• Cellular telecommunications
• Subset sums and Cayley graphs