Linear inequalities for covering codes: Part I: Pair covering inequalities

The lower bounds on K(n,R), minimum number of codewords of any binary code of length n, and covering radius R are improved. A new technique combining the Hamming association scheme and the results of a classic problem of covering pairs by k-tuples is introduced. The new lower bounds are obtained by studying various linear inequalities satisfied by covering codes, and such an inequality is derived. The lower bounds for K(n,R) are studied for some special values of n and R, which serve as examples showing how the methods developed are applied to concrete cases. >

[1]  Zhen Zhang,et al.  Lower bounds on t[n, k] from linear inequalities , 1992, IEEE Trans. Inf. Theory.

[2]  R. G. Stanton,et al.  Maximal and minimal coverings of (k − 1)-tuples by k-tuples , 1968 .

[3]  R. G. Stanton,et al.  A Combinatorial Problem in Matching , 1969 .

[4]  Gérard D. Cohen,et al.  Covering radius - Survey and recent results , 1985, IEEE Trans. Inf. Theory.

[5]  Patrick Solé.  A Lloyd Theorem in Weakly Metric Association Schemes , 1989, Eur. J. Comb..

[6]  N. J. A. Sloane,et al.  Further results on the covering radius of codes , 1986, IEEE Trans. Inf. Theory.

[7]  Ewald W. Weber.  On the Football Pool Problem for 6 Matches: A New Upper Bound , 1983, J. Comb. Theory, Ser. A.

[8]  Marion Fort,et al.  Minimal coverings of pairs by triples , 1958 .

[9]  J. G. Mauldon.  Covering theorems for groups , 1950 .

[10]  N. J. A. Sloane,et al.  Inequalities for covering codes , 1988, IEEE Trans. Inf. Theory.

[11]  R. G. Stanton,et al.  Covering Theorems for Vectors with Special Reference to the Case of Four and Five Components , 1969 .

[12]  Juriaan Simonis,et al.  The minimal covering radius t[15, 6] of a six-dimensional binary linear code of length 15 is equal to 4 , 1988, IEEE Trans. Inf. Theory.

[13]  F. MacWilliams,et al.  The Theory of Error-Correcting Codes , 1977 .

[14]  Hiram Fernandes,et al.  The Football Pool Problem for 7 and 8 Matches , 1983, J. Comb. Theory, Ser. A.

[15]  N. J. A. Sloane,et al.  On the covering radius of codes , 1985, IEEE Trans. Inf. Theory.

[16]  Iiro Honkala,et al.  A new construction for covering codes , 1988, IEEE Trans. Inf. Theory.

[17]  Joseph Douglas Horton,et al.  On Covering Sets and Error-Correcting Codes , 1971, J. Comb. Theory, Ser. A.

[18]  Gerhard J. M. van Wee,et al.  Improved sphere bounds on the coveting radius of codes , 1988, IEEE Trans. Inf. Theory.

[19]  Gerald Losey.  Note on a theorem of Zaremba , 1969 .

[20]  Haim Hanani,et al.  On Quadruple Systems , 1960, Canadian Journal of Mathematics.

[21]  T. J. Dickson.  On a Covering Problem Concerning Abelian Groups , 1971 .

[22]  Iiro S. Honkala,et al.  Modified bounds for coveting codes , 1991, IEEE Trans. Inf. Theory.

[23]  S. K. Zaremba.  A Covering Theorem for Abelian Groups , 1951 .

[24]  S. K. Zaremba.  Covering Problems Concerning Abelian Groups , 1952 .

[25]  R. G. Stanton,et al.  Covering Problems for Dichotomized Matching , 1968 .

[26]  R. G. Stanton,et al.  COVERINGS OF PAIRS BY k‐SETS , 1970 .

[27]  J. H. van Lint,et al.  The football pool problem for 5 matches , 1967 .