Covering code

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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[edit]

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 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.

Example[edit]

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[edit]

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

Football pools problem[edit]

A particular case is the football pools problem, based on football pool betting, where the aim is to predict the results of n football matches as a home win, draw or away win, or to at least predict n - 1 of them with multiple bets. Thus a ternary covering, K3(n,1), is sought.

If n=\tfrac12 (3^k-1) then 3n-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

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14
K3(n,1) 1 3 5 9 27 71-73 156-186 402-486 1060-1269 2854-3645 7832-9477 21531-27702 59049 166610-177147
K3(n,2) 1 3 3 8 15-17 26-34 54-81 130-219 323-555 729 1919-2187 5062-6561 12204-19683
K3(n,3) 1 3 3 6 11-12 14-27 27-54 57-105 117-243 282-657 612-1215 1553-2187

Applications[edit]

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

References[edit]

  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. ^ http://alexandria.tue.nl/repository/freearticles/593454.pdf
  4. ^ http://www.sztaki.hu/~keri/codes/3_tables.pdf
  5. ^ G. Cohen, I. Honkala, S. Litsyn, A. Lobstein, Covering Codes, Elsevier (1997) ISBN 0-444-82511-8
  6. ^ 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[edit]