Abstract
As a generalization to cyclic and consta-cyclic codes, quasi-twisted (QT) codes contain many good linear codes. During the last twenty years, a lot of record-breaking codes have been found by computer search for good QT codes. But due to the time complexity, very few QT codes have been reported recently. In this paper, a new iterative, heuristic computer search algorithm is presented, and a lot of new QT codes have been obtained. With these results, a total of 45 entries in the code tables for the best-known codes have been improved. Also, as an example to show the effectiveness of the algorithm, 8 better binary quasi-cyclic codes with dimension 12 and \(m = 13\) than previously best-known results are constructed.
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Aydin N., Siap I.: New quasi-cyclic codes over F\(_{5}\). Appl. Math. Lett. 15, 833–836 (2002).
Aydin N., Siap I., Ray-Chaudhuri K.: The structure of 1-generator quasi-twisted codes and new linear codes. Des. Codes Cryptogr. 24, 313–326 (2001).
Berlekamp E.R.: Algebraic Coding Theory. Aegean Park Press, Laguna Hills (1984).
Bhargava V.K., Seguin G.E., Stein J.M.: Some (mk, k) cyclic codes in quasi-cyclic form. IEEE Trans. Inf. Theory 24, 358–369 (1978).
Bosma W., Cannon J., Playoust C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24, 235–265 (1997).
Calderbank A.R., Kantor W.M.: The geometry of two-weight codes. Bull. Lond. Math. Soc. 18, 97–122 (1986).
Chen E.Z.: Six new binary quasi-cyclic codes. IEEE Trans. Inf. Theory 40, 1666–1667 (1994).
Chen E.Z.: Quasi-cyclic codes derived from cyclic codes. In: Proceedings of the International Symposium on Information Theory and its Applications (ISITA2004), Parma, pp. 162–165 (2004).
Chen E.Z.: New quasi-cyclic codes from simplex codes. IEEE Trans. Inf. Theory 53, 1193–1196 (2007).
Chen E.Z.: Web Database of Binary QC Codes. http://moodle.tec.hkr.se/~chen/research/codes/searchqc2.htm. Accessed 11 Mar 2014.
Chen C.L., Peterson W.W.: Some results on quasi-cyclic codes. Inf. Control 15, 407–423 (1969).
Daskalov R., Hristov P.: New quasi-twisted degenerate ternary linear codes. IEEE Trans. Inf. Theory 49, 2259–2263 (2003).
Daskalov R.N., Gulliver T.A., Metodieva E.: New good quasi-cyclic ternary and quaternary linear codes. IEEE Trans. Inf. Theory 43, 1647–1650 (1997).
Daskalov R.N., Gulliver T.A., Metodieva E.: New ternary linear codes. IEEE Trans. Inf. Theory 45, 1687–1688 (1999).
Daskalov R., Hristov P., Metodieva E.: New minimum distance bounds for linear codes over GF(5). Discret. Math. 275, 97–110 (2004).
Grassl M.: Tables of linear codes and quantum codes. http://codetables.de. Accessed 11 Mar 2014.
Greenough P.P., Hill R.: Optimal ternary quasi-cyclic codes. Des. Codes Cryptogr. 2, 81–91 (1992).
Gulliver T.A.: New optimal ternary linear codes. IEEE Trans. Inf. Theory 41, 1182–1185 (1995).
Gulliver T.A., Bhargava V.K.: Some best rate 1/p and rate (p \(-\) 1)/p systematic quasi-cyclic codes. IEEE Trans. Inf. Theory 37, 552–555 (1991).
Gulliver T.A., Bhargava V.K.: Nine good (m \(-\) 1)/pm quasi-cyclic codes. IEEE Trans. Inf. Theory 38, 1366–1369 (1992).
Gulliver T.A., Bhargava V.K.: some best rate 1/p and rate (p \(-\) 1)/p systematic quasi-cyclic codes over GF(3) and GF(4). IEEE Trans. Inf. Theory 38, 1369–1374 (1992).
Gulliver T.A., Bhargava V.K.: New good rate (m \(-\) 1)/pm ternary and quaternary quasi-cyclic codes. Des. Codes Cryptogr. 7, 223–233 (1996).
Gulliver T.A., Östergård J.: Improved bounds for ternary linear codes of dimension 7. IEEE Trans. Inf. Theory 43, 1377–1388 (1997).
Gulliver T.A., Östergård P.R.J.: Improved bounds for quaternary linear codes of dimension 6. Appl. Algebra Eng. Commun. Comput. 9, 153–159 (1998).
Gulliver T.A., Östergård P.R.J.: New binary linear codes. Ars Comb. 56, 105–112 (2000).
Gulliver T.A., Östergård P.R.J.: Improved bounds for ternary linear codes of dimension 8 using tabu search. J. Heuristics 7, 37–46 (2000).
Heijnen P., vanTiborg H.C.A., Verhoeff T., Weijs S.: Some new binary quasi-cyclic codes. IEEE Trans. Inf. Theory 44, 1994–1996 (1998).
Kasami T.: A gilbert-varshamov bound for quasi-cyclic codes of rate 1/2. IEEE Trans. Inf. Theory 20, 679 (1974).
Maruta T., et. al.: New linear codes from cyclic or generalized cyclic codes by puncturing. In: Proceedings of the 10th International Workshop on Algebraic and Combinatorial Coding Theory (ACCT-10), Zvenigorod, pp. 194–197 (2006).
Siap I., Aydin N., Ray-Chaudhuri D.K.: New ternary quasi-cyclic codes with better minimum distances. IEEE Trans. Inf. Theory 46, 1554–1558 (2000).
Townsend R.L., Weldon E.: Self-orthogonal quasi-cyclic codes. IEEE Trans. Inf. Theory 13, 183–195 (1967).
Van Tilborg H.C.A.: On quasi-cyclic codes with rate 1/m. IEEE Trans. Inf. Theory 24, 628–629 (1978).
Vardy A.: The intractability of computing the minimum distance of a code. IEEE Trans. Inf. Theory 43, 1757–1766 (1997).
Venkaiah V.C., Gulliver T.A.: Quasi-cyclic codes over F\(_{13}\) and enumeration of defining polynomials. J. Discret. Algorithms 16, 249–257 (2012).
Weldon E.J. Jr.: Long quasi-cyclic codes are good. IEEE Trans. Inf. Theory 16, 130 (1970).
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The author is grateful to the referees for their helpful comments and suggestions that significantly improved the presentation of the results.
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Communicated by R. Hill.
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Chen, E.Z. A new iterative computer search algorithm for good quasi-twisted codes. Des. Codes Cryptogr. 76, 307–323 (2015). https://doi.org/10.1007/s10623-014-9950-8
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DOI: https://doi.org/10.1007/s10623-014-9950-8