Abstract
Previous results have shown that the class of quasi-cyclic (QC) codes contains many good codes. In this paper, new rate (m-1)/pm QC codes over GF(3) and GF(4) are presented. These codes have been constructed using integer linear programming and a heuristic combinatorial optimization algorithm based on a greedy local search. Most of these codes attain the maximum possible minimum distance for any linear code with the same parameters, i.e., they are optimal, and 58 improve the maximum known distances. The generator polynomials for these 58 codes are tabulated, and the minimum distances of rate (m-1)/pm QC codes are given.
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References
E. H. L. Aarts and P. J. M. van Laarhoven, Local search in coding theory, Discrete Math., Vol. 106/107 (1992) pp. 11–18.
A. E. Brouwer, Table of minimum-distance bounds for linear codes over GF(3) and GF(4), lincodbd server, aeb@cwi.nl, Eindhoven University of Technology, Eindhoven, the Netherlands (1993).
R. N. Daskalov, R. Hill and P. Lizak, Tables of bounds on linear codes over GF(3) and GF(4), Technical University, Gabrovo, Bulgaria (1992).
H. Fredricksen and J. Maiorana, Necklaces of beads in k colors and k-ary de Bruijn sequences, Discrete Math., Vol. 23 (1978) pp. 207–210.
R. G. Gallager, The random coding bound is tight for the average code, IEEE Trans. Inf. Theory, Vol. IT-19 (1973) pp. 244–246.
P. P. Greenough and R. Hill, Optimal ternary quasi-cyclic codes, Designs, Codes and Cryptography, Vol. 2 (1992) pp. 81–91.
T. A. Gulliver and V. K. Bhargava, Some best rate 1/p and rate (p − 1)/p systematic quasi-cyclic codes, IEEE Trans. Inf. Theory, Vol. IT-37 (1991) pp. 552–555.
T. A. Gulliver and V. K. Bhargava, Some best rate 1/p and rate (p − 1)/p systematic quasi-cyclic codes over GF(3) and GF(4), IEEE Trans. Inf. Theory, Vol. IT-38 (1992) pp. 1369–1374.
T. A. Gulliver and V. K. Bhargava, Nine good rate (m − 1)/pm quasi-cyclic codes, IEEE Trans. Inf. Theory, Vol. IT-38 (1992) pp. 1366–1369.
T. A. Gulliver and V. K, Bhargava, V.K. Twelve good rate (m − r)/pm quasi-cyclic codes, IEEE Trans. Inf. Theory, Vol. IT-39 (1993).
M. HallJr., Combinatorial Theory, Blaisdell Publishing Co., Waltham, MA (1967).
A. A. Hashim and A. G. Constantinides, Some new results on binary linear block codes, Electronics Letters, Vol. 10 (1974) pp. 31–33.
A. A. Hashim and V. S. Pozdniakov, Computerized search for linear binary codes. Electronics Letters, Vol. 12 (1976) pp. 350–351.
T. Kasami, A Gilbert-Varshamov bound for quasi-cyclic codes of rate 1/2, IEEE Trans. Inf. Theory, Vol. IT- 20 (1974) pp. 679.
F. R. Kschischang and S. Pasupathy, Some ternary and quaternary codes and associated sphere packings, IEEE Trans. Inf. Theory, Vol. IT-38 (1992) pp. 227–246.
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland Publishing Co., New York (1977).
J. N. Pierce, Limit distribution of the minimum distance of random linear codes, IEEE Trans. Inf. Theory, Vol. IT-13 (1967) pp. 595–599.
G. E. Séguin and G. Drolet, The Theory of 1-Generator Quasi-Cyclic Codes, Royal Military College of Canada, Kingston, ON (1991).
N. J. A. Sloane, Tables of lower bounds on dmax(n,k) for linear codes over fields of order 3 and 4, to appear in V. Pless, et al., The Handbook of Coding Theory.
G. Solomon and J. J. Stiffler, Algebraically punctured cyclic codes, Inf. and Contr., Vol. 8 (1965) pp. 170–179.
H. C. A. van Tilborg, On quasi-cyclic codes with rate 1/m. IEEE Trans. Inf. Theory, Vol. IT-24 (1978) pp. 628–629.
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Communicated by: R. Mullin
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Gulliver, T.A., Bhargava, V.K. New good rate (m-1)/pm ternary and quaternary quasi-cyclic codes. Des Codes Crypt 7, 223–233 (1996). https://doi.org/10.1007/BF00124513
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DOI: https://doi.org/10.1007/BF00124513