Abstract
Let K(n,r) denote the minimum cardinality of a binary covering code of length n and covering radius r. Constructions of covering codes give upper bounds on K(n,r). It is here shown how computer searches for covering codes can be sped up by requiring that the code has a given (not necessarily full) automorphism group. Tabu search is used to find orbits of words that lead to a covering. In particular, a code D found which proves K(13,1) ≤ 704, a new record. A direct construction of D given, and its full automorphism group is shown to be the general linear group GL(3,3). It is proved that D is a perfect dominating set (each word not in D is covered by exactly one word in D) and is a counterexample to the recent Uniformity Conjecture of Weichsel.
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References
A. Blokhuis and C. W. H. Lam, More coverings by rook domains, J. Combin. Theory Ser. A, Vol. 36 (1984) pp. 240-244.
G. D. Cohen, A. C. Lobstein, and N. J. A. Sloane, Further results on the covering radius of codes, IEEE Trans. Inform. Theory, Vol. 32 (1986) pp. 680-694.
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford (1985).
F. Glover, Tabu search—Part I, ORSA J. Comput., Vol. 1 (1989) pp. 190-206.
M. Hall, Jr., Combinatorial Theory, Blaisdell, Waltham, MA (1967).
I. S. Honkala and P. R. J. Östergård, Code design, Local Search in Combinatorial Optimization (E. Aarts and J. K. Lenstra, eds.), Wiley, New York (1997) pp. 441-456.
P. J. M. van Laarhoven, E. H. L. Aarts, J. H. van Lint, and L. T. Wille, New upper bounds for the football pool problem for 6, 7, and 8 matches, J. Combin. Theory Ser. A, Vol. 52 (1989) pp. 304-312.
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam (1977).
P. R. J. Östergård, New upper bounds for the football pool problem for 11 and 12 matches, J. Combin. Theory Ser. A, Vol. 67 (1994) pp. 161-168.
P. R. J. Östergård, A combinatorial proof for the football pool problem for six matches, J. Combin. Theory Ser. A, Vol. 76 (1996) pp. 160-163.
P. R. J. Östergård, Constructing covering codes by tabu search, J. Combin. Des., Vol. 5 (1997) pp. 71-80.
P. R. J. Östergård, New constructions for q-ary covering codes, Ars Combin., to appear.
P. R. J. Östergård and H. O. Hämäläinen, A new table of binary/ternary mixed covering codes, Des. Codes Cryptogr., Vol. 11 (1997) pp. 151-178.
P. R. J. Östergård and M. K. Kaikkonen, New upper bounds for binary covering codes, Discrete Math., Vol. 178 (1998) pp. 165-179.
R. G. Stanton and J. G. Kalbfleisch, Covering problems for dichotomized matchings, Aequationes Math., Vol. 1 (1968) pp. 94-103.
W. D. Weakley, Minimal dominating sets in hypercubes, Proceedings of the Eighth International Conference on Graph Theory, Combinatorics, Algorithms, and Applications, to appear.
P. M. Weichsel, Dominating sets in n-cubes, J. Graph Theory, Vol. 18 (1994) pp. 479-488.
L. T. Wille, The football pool problem for 6 matches: a new upper bound obtained by simulated annealing, J. Combin. Theory Ser. A, Vol. 45 (1987) pp. 171-177.
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Osterg, P.R.J., Weakley, W.D. Constructing Covering Codes with Given Automorphisms. Designs, Codes and Cryptography 16, 65–73 (1999). https://doi.org/10.1023/A:1008326409439
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DOI: https://doi.org/10.1023/A:1008326409439