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Designs, Codes and Cryptography

, Volume 72, Issue 3, pp 749–763 | Cite as

Lexicodes over rings

  • Kenza Guenda
  • T. Aaron Gulliver
  • S. Arash Sheikholeslam
Article

Abstract

In this paper, we consider the construction of linear lexicodes over finite chain rings by using a \(B\)-ordering over these rings and a selection criterion. As examples we give lexicodes over \(\mathbb Z _4\) and \(\mathbb F _2+u\mathbb F _2\). It is shown that this construction produces many optimal codes over rings and also good binary codes. Some of these codes meet the Gilbert bound. We also obtain optimal self-dual codes, in particular the octacode.

Keywords

Codes over rings Greedy codes Self-orthogonal codes Self-dual codes Gray map 

Mathematics Subject Classification

94B65 94B05 16T99 

Notes

Acknowledgments

The authors would like to thank the reviewers for their useful comments which improved the paper considerably. In addition, we would like to thank the Editor for their careful handling of our paper.

References

  1. 1.
    Aoki T., Gaborit P., Harada M., Ozeki M., Solé P.: On the covering radius of \(\mathbb{Z}_4\)-codes and their lattices. IEEE Trans. Inf. Theory 45(6), 2162–2168 (1999).Google Scholar
  2. 2.
    Aydin N., Asamov T.: Database for codes over \(\mathbb{Z}_4\). Available online at: http://www.asamov.com/Z4Codes/CODES/ShowCODESTablePage.aspx.
  3. 3.
    Bonn J.T.: Forcing linearity on greedy codes. Des. Codes Cryptogr. 9(1), 39–49 (1996).Google Scholar
  4. 4.
    Brualdi R.A., Pless V.: Greedy codes. J. Comb. Theory Ser. A 64, 10–30 (1993).Google Scholar
  5. 5.
    Conway J.H., Sloane N.J.A.: Lexicographic codes: error-correcting codes from game theory. IEEE Trans. Inf. Theory 32(3), 337–348 (1986).Google Scholar
  6. 6.
    Dougherty S.T., Gaborit P., Harada M., Solé P.: Type II codes over \(\mathbb{F}_2+u\mathbb{F}_2\). IEEE Trans. Inf. Theory, 45(1), 32–45 (1999).Google Scholar
  7. 7.
    Dougherty S.T., Gulliver A., Park Y.H.: Optimal linear codes over \(\mathbb{Z}_m\). J. Korean Math. Soc. 44(5), 1139–1162 (2007).Google Scholar
  8. 8.
    Grassl M.: Bounds on the minimum distance of linear codes and quantum codes. Available online at: http://www.codetables.de.
  9. 9.
    Greferath M., Schmidt S.E.: Gray isometries for finite chain rings and non-linear ternary \((36,3^{12},15)\). IEEE. Trans. Inf. Theory 45(7), 2522–2524 (1999).Google Scholar
  10. 10.
    Guenda K.: Gulliver T.A.: MDS and self-dual codes over rings. Finite Fields Appl. 18(6), 1061–1075 (2012).Google Scholar
  11. 11.
    Hammons A.R. Jr., Kumar P.V., Calderbank A.R., Sloane N.J.A., Solé P.: The \(\mathbb{Z}_4\)-linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inf. Theory 40(2), 301–319 (1994).Google Scholar
  12. 12.
    Honold T., Nechaev A.A.: Weighted modules and representations of codes. Tech. Univ. München, Fak. Math. Report, Beitrage Zur Geometrie and Algebra 36 (1998).Google Scholar
  13. 13.
    Huffman W.C., Pless V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, New York (2003).Google Scholar
  14. 14.
    Levenstein T.: A class of systematic codes. Soviet Math. Dokl. 1, 368–371 (1960)Google Scholar
  15. 15.
    van Zanten A.J.: Lexicographic order and linearity. Des. Codes Cryptogr. 10(1), 85–97 (1997).Google Scholar
  16. 16.
    van Zanten A.J., Nengah Suparta I.: On the construction of linear \(q\)-ary lexicodes. Des. Codes Cryptogr. 37(1), 15–29 (2005).Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Kenza Guenda
    • 1
  • T. Aaron Gulliver
    • 1
  • S. Arash Sheikholeslam
    • 1
  1. 1.Department of Electrical and Computer EngineeringUniversity of VictoriaVictoriaCanada

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