Quasi-Cyclic Codes over \(\mathbb{F}_{13}\)

  • T. Aaron Gulliver
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7056)


Let d q (n,k) be the maximum possible minimum Hamming distance of a linear [n,k] code over \(\mathbb{F}_{q}\). Tables of best known linear codes exist for all fields up to q = 9. In this paper, linear codes over \(\mathbb{F}_{13}\) are constructed for k up to 6. The codes constructed are from the class of quasi-cyclic codes. In addition, the minimum distance of the extended quadratic residue code of length 44 is determined.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Berlekamp, E.R.: Algebraic Coding Theory. McGraw-Hill, New York (1968)zbMATHGoogle Scholar
  2. 2.
    Betsumiya, K., Georgiou, S., Gulliver, T.A., Harada, M., Koukouvinos, C.: On self-dual codes over some prime fields. Disc. Math. 262, 37–58 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bosma, W., Cannon, J.: Handbook of Magma Functions, Department of Mathematics, University of Sydney,
  4. 4.
    Daskalov, R.N., Gulliver, T.A.: New good quasi-cyclic ternary and quaternary linear codes. IEEE Trans. Inform. Theory 43, 1647–1650 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Glover, F.: Tabu search—Part I. ORSA J. Comput. 1, 190–206 (1989)CrossRefzbMATHGoogle Scholar
  6. 6.
    Grassl, M.: Bounds on the minimum distance of linear codes and quantum codes,
  7. 7.
    de Boer, M.A.: Almost MDS codes. Design, Codes Crypt. 9, 143–155 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Greenough, P.P., Hill, R.: Optimal ternary quasi-cyclic codes. Designs, Codes and Crypt. 2, 81–91 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    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. Inform. Theory 38, 1369–1374 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gulliver, T.A.: New optimal ternary linear codes. IEEE Trans. Inform. Theory 41, 1182–1185 (1995)CrossRefzbMATHGoogle Scholar
  11. 11.
    Gulliver, T.A., Bhargava, V.K.: New good rate (m-1)/pm ternary and quaternary quasi-cyclic codes. Designs, Codes and Crypt. 7, 223–233 (1996)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Honkala, I.S., Östergård, P.R.J.: Applications in code design. In: Aarts, E.H.L., Lenstra, J.K. (eds.) Local Search in Combinatorial Optimization. Wiley, New York (2003)Google Scholar
  13. 13.
    MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland, New York (1977)zbMATHGoogle Scholar
  14. 14.
    Newhart, D.W.: On minimum weight codewords in QR codes. J. Combin. Theory Ser. A 48, 104–119 (1988)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • T. Aaron Gulliver
    • 1
  1. 1.Department of Electrical and Computer EngineeringUniversity of VictoriaVictoriaCanada

Personalised recommendations