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Designs, Codes and Cryptography

, Volume 76, Issue 2, pp 307–323 | Cite as

A new iterative computer search algorithm for good quasi-twisted codes

  • Eric Zhi ChenEmail author
Article

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.

Keywords

Best-known codes Coding theory Search algorithm Linear codes Quasi-twisted codes Simplex code 

Mathematics Subject Classification

94B05 

Notes

Acknowledgments

The author is grateful to the referees for their helpful comments and suggestions that significantly improved the presentation of the results.

References

  1. 1.
    Aydin N., Siap I.: New quasi-cyclic codes over F\(_{5}\). Appl. Math. Lett. 15, 833–836 (2002).Google Scholar
  2. 2.
    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).Google Scholar
  3. 3.
    Berlekamp E.R.: Algebraic Coding Theory. Aegean Park Press, Laguna Hills (1984).Google Scholar
  4. 4.
    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).Google Scholar
  5. 5.
    Bosma W., Cannon J., Playoust C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24, 235–265 (1997).Google Scholar
  6. 6.
    Calderbank A.R., Kantor W.M.: The geometry of two-weight codes. Bull. Lond. Math. Soc. 18, 97–122 (1986).Google Scholar
  7. 7.
    Chen E.Z.: Six new binary quasi-cyclic codes. IEEE Trans. Inf. Theory 40, 1666–1667 (1994).Google Scholar
  8. 8.
    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).Google Scholar
  9. 9.
    Chen E.Z.: New quasi-cyclic codes from simplex codes. IEEE Trans. Inf. Theory 53, 1193–1196 (2007).Google Scholar
  10. 10.
    Chen E.Z.: Web Database of Binary QC Codes. http://moodle.tec.hkr.se/~chen/research/codes/searchqc2.htm. Accessed 11 Mar 2014.
  11. 11.
    Chen C.L., Peterson W.W.: Some results on quasi-cyclic codes. Inf. Control 15, 407–423 (1969).Google Scholar
  12. 12.
    Daskalov R., Hristov P.: New quasi-twisted degenerate ternary linear codes. IEEE Trans. Inf. Theory 49, 2259–2263 (2003).Google Scholar
  13. 13.
    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).Google Scholar
  14. 14.
    Daskalov R.N., Gulliver T.A., Metodieva E.: New ternary linear codes. IEEE Trans. Inf. Theory 45, 1687–1688 (1999).Google Scholar
  15. 15.
    Daskalov R., Hristov P., Metodieva E.: New minimum distance bounds for linear codes over GF(5). Discret. Math. 275, 97–110 (2004).Google Scholar
  16. 16.
    Grassl M.: Tables of linear codes and quantum codes. http://codetables.de. Accessed 11 Mar 2014.
  17. 17.
    Greenough P.P., Hill R.: Optimal ternary quasi-cyclic codes. Des. Codes Cryptogr. 2, 81–91 (1992).Google Scholar
  18. 18.
    Gulliver T.A.: New optimal ternary linear codes. IEEE Trans. Inf. Theory 41, 1182–1185 (1995).Google Scholar
  19. 19.
    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).Google Scholar
  20. 20.
    Gulliver T.A., Bhargava V.K.: Nine good (m \(-\) 1)/pm quasi-cyclic codes. IEEE Trans. Inf. Theory 38, 1366–1369 (1992).Google Scholar
  21. 21.
    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).Google Scholar
  22. 22.
    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).Google Scholar
  23. 23.
    Gulliver T.A., Östergård J.: Improved bounds for ternary linear codes of dimension 7. IEEE Trans. Inf. Theory 43, 1377–1388 (1997).Google Scholar
  24. 24.
    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).Google Scholar
  25. 25.
    Gulliver T.A., Östergård P.R.J.: New binary linear codes. Ars Comb. 56, 105–112 (2000).Google Scholar
  26. 26.
    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).Google Scholar
  27. 27.
    Heijnen P., vanTiborg H.C.A., Verhoeff T., Weijs S.: Some new binary quasi-cyclic codes. IEEE Trans. Inf. Theory 44, 1994–1996 (1998).Google Scholar
  28. 28.
    Kasami T.: A gilbert-varshamov bound for quasi-cyclic codes of rate 1/2. IEEE Trans. Inf. Theory 20, 679 (1974).Google Scholar
  29. 29.
    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).Google Scholar
  30. 30.
    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).Google Scholar
  31. 31.
    Townsend R.L., Weldon E.: Self-orthogonal quasi-cyclic codes. IEEE Trans. Inf. Theory 13, 183–195 (1967).Google Scholar
  32. 32.
    Van Tilborg H.C.A.: On quasi-cyclic codes with rate 1/m. IEEE Trans. Inf. Theory 24, 628–629 (1978).Google Scholar
  33. 33.
    Vardy A.: The intractability of computing the minimum distance of a code. IEEE Trans. Inf. Theory 43, 1757–1766 (1997).Google Scholar
  34. 34.
    Venkaiah V.C., Gulliver T.A.: Quasi-cyclic codes over F\(_{13}\) and enumeration of defining polynomials. J. Discret. Algorithms 16, 249–257 (2012).Google Scholar
  35. 35.
    Weldon E.J. Jr.: Long quasi-cyclic codes are good. IEEE Trans. Inf. Theory 16, 130 (1970).Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Computer ScienceKristianstad UniversityKristianstadSweden

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