Skip to main content
SpringerLink
Log in

New binary self-dual codes of lengths 50–60

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

Using a method for constructing binary self-dual codes with an automorphism of odd prime order \(p\), we give a full classification of all optimal binary self-dual \([50+2t,25+t]\) codes having an automorphism of order 5 for \(t=0,\dots ,5\). As a consequence, we determine the weight enumerators for which there is an optimal binary self-dual \([52, 26, 10]\) code. Some of the constructed codes for lengths 52, 54, 58, and 60 have new values for the parameter in their weight enumerator. We also construct more than 3,000 new doubly-even \([56,28,12]\) self-dual codes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Bouyukliev I.: About the code equivalence, Advances in Coding Theory and Cryptography. Series on Coding Theory and Cryptology, vol. 3, pp. 126–151. World Scientific Publishing (2007).

  2. Bouyuklieva S.: New extremal self-dual codes of lengths 42 and 44. IEEE Trans. Inf. Theory 43, 1607–1612 (1997).

    Google Scholar 

  3. Bouyuklieva S.: Some optimal self-orthogonal and self-dual codes. Discret. Math. 287, 1–10 (2004).

    Google Scholar 

  4. Bouyuklieva S., Harada M., Munemasa A.: Restrictions on the weight enumerators of binary self-dual codes of length 4 m. In: Proc. International Workshop Optimal Codes and Related Topics, (OCRT’07), White Lagoon, pp. 40–44 (2007).

  5. Bouyuklieva S., Malevich A., Willems W.: Automorphisms of extremal self-dual codes. IEEE Trans. Inf. Theory 56, 2091–2096 (2010).

    Google Scholar 

  6. Bouyuklieva S., Östergård P.: New constructions of optimal self-dual binary codes of length 54. Des. Codes Cryptogr. 41, 101–109 (2006).

    Google Scholar 

  7. Bouyuklieva S., Yankov N., Kim J.L.: Classification of binary self-dual \([48, 24, 10]\) codes with an automorphism of odd prime order. Finite Fields Appl. 18(6), 1104–1113 (2012).

    Google Scholar 

  8. Conway J.H., Sloane N.J.A.: A new upper bound on the minimal distance of self-dual codes. IEEE Trans. Inf. Theory 36, 1319–1333 (1990).

    Google Scholar 

  9. Harada M.: Self-orthogonal \(3-(56, 12, 65)\) designs and extremal doubly-even self-dual codes of length 56. Des. Codes Cryptogr. 38, 5–16 (2006).

  10. Harada M., Gulliver T.A., Kaneta H.: Classification of extremal double circulant self-dual codes of length up to 62. Discret. Math. 188, 127–136 (1998).

    Google Scholar 

  11. Harada M., Munemasa A.: Some restrictions on weight enumerators of singly even self-dual codes. IEEE Trans. Inf. Theory 52, 1266–1269 (2006).

    Google Scholar 

  12. Harada M., Munemasa A., Tonchev V.: A characterization of designs related to an extremal doubly-even self-dual code of length 48. Ann. Comb. 9, 189–198 (2005).

    Google Scholar 

  13. Huffman W.C.: Automorphisms of codes with application to extremal doubly-even codes of length 48. IEEE Trans. Inf. Theory 28, 511–521 (1982).

    Google Scholar 

  14. Huffman W.C.: On the classification and enumeration of self-dual codes. Finite Fields Appl. 11, 451–490 (2005).

    Google Scholar 

  15. Huffman W.C., Pless V.: Fundamentals of error correcting codes. Cambridge University Press, Cambridge (2003).

  16. Pless V.: A classification of self-orthogonal codes over GF(2). Discret. Math. 3, 209–246 (1972).

    Google Scholar 

  17. Rains E.M.: Shadow bounds for self-dual codes. IEEE Trans. Inf. Theory 44, 134–138 (1998).

    Google Scholar 

  18. Yankov N.: New optimal [52, 26, 10] self-dual codes. Des. Codes Cryptogr. (2012). doi:10.1007/s10623-012-9639-9.

  19. Yankov N., Russeva R.: Binary self-dual codes of lengths 52 to 60 with an automorphism of order 7 or 13. IEEE Trans. Inf. Theory 56, 7498–7506 (2011).

    Google Scholar 

  20. Yorgov V.: Binary self-dual codes with an automorphism of odd order. Probl. Inf. Transm. 19, 260–270 (1983).

    Google Scholar 

  21. Yorgov V.: A method for constructing inequivalent self-dual codes with applications to length 56. IEEE Trans. Inf. Theory 33, 77–82 (1987).

    Google Scholar 

Download references

Acknowledgments

We thank the anonymous referees for the helpful comments and suggestions, which significantly contributed to improving the quality of the publication. This paper was studied with the support of the Ministry of Education Science and Technology (MEST) and the Korean Federation of Science and Technology Societies (KOFST). This work was also supported by World Class University Project (WCU) R-32-2012-000-20014-0, Basic Science Research Program 2010-0020942, NRF Korea, and MEST 2012-002521, NRF Korea. The first author was also supported by Shumen University under Grant RD-08-245/13.03.2013.

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Informatics, Shumen University, 9700 , Shumen, Bulgaria

    Nikolay Yankov

  2. Division of Electronics & Information Engineering, Chonbuk National University, 664-14, 1GA Dekjin-Dong, Jeonju City, Jeonbuk, 561-756, South Korea

    Moon Ho Lee

Authors
  1. Nikolay Yankov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Moon Ho Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Moon Ho Lee.

Additional information

Communicated by D. Jungnickel.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yankov, N., Lee, M.H. New binary self-dual codes of lengths 50–60. Des. Codes Cryptogr. 73, 983–996 (2014). https://doi.org/10.1007/s10623-013-9839-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-013-9839-y

Keywords

Mathematics Subject Classification

Search

Navigation