Covering code

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.

Let q ≥ 2 {\displaystyle q\geq 2} , n ≥ 1 {\displaystyle n\geq 1} , R ≥ 0 {\displaystyle R\geq 0} be integers. A code C ⊆ Q n {\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 y ∈ Q n {\displaystyle y\in Q^{n}} there is a codeword x ∈ C {\displaystyle x\in C} such that the Hamming distance d H ( x , y ) ≤ R {\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 Q n {\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 K q ( n , R ) {\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 K q ( n , 1 ) ≥ q n − 1 / ( n − 1 ) {\displaystyle K_{q}(n,1)\geq q^{n-1}/(n-1)} and K q ( n , n − 2 ) ≥ q 2 / ( n − 1 ) {\displaystyle K_{q}(n,n-2)\geq q^{2}/(n-1)} .^[2] The covering problem is closely related to the packing problem in Q n {\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 n = 1 2 ( 3 k − 1 ) {\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

• Literature on covering codes
• Bounds on K q ( n , R ) {\displaystyle K_{q}(n,R)}