On the automorphisms of order 15 for a binary self-dual $$[96, 48, 20]$$ code | Designs, Codes and Cryptography Skip to main content
Springer Nature Link
Log in

On the automorphisms of order 15 for a binary self-dual \([96, 48, 20]\) code

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

Abstract

The structure of the binary self-dual codes invariant under the action of a cyclic group of order \(pq\) for odd primes \(p\ne q\) is considered. As an application we prove the nonexistence of an extremal self-dual \([96, 48, 20]\) code with an automorphism of order \(15\) which closes a gap in de la Cruz and Willems (IEEE Trans Inf Theory 57:6820–6823, 2011).

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Japan)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Assmus E.F., Mattson H.F.: New 5-designs. J. Comb. Theory 6, 122–151 (1969).

  2. Bouyukliev I.: What is Q-extension? Serdica J. Comput. 1, 115–130 (2007).

  3. de la Cruz J., Willems W.: On extremal self-dual codes of length 96. IEEE Trans. Inf. Theory 57, 6820–6823 (2011).

  4. Grassl M.: Bounds on the minimum distance of linear codes and quantum codes. http://www.codetables.de.

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

  6. Huffman W.C.: Decomposing and shortening codes using automorphisms. IEEE Trans. Inf. Theory 32, 833–836 (1986).

  7. Huffman W.C., Pless V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003).

  8. Kéri G.: Tables for \(n\)-arcs and lists for complete \(n\)-arcs in \(PG(r, q)\). http://www.sztaki.hu/~keri/n-arcs/index.htm.

  9. Ling S., Solé P.: On the algebraic structure of quasi-cyclic codes I: Finite fields. IEEE Trans. Inf. Theory 47, 2751–2760 (2001).

  10. Rains E.M., Sloane N.J.A.: Self-dual codes. In: Pless V.S., Huffman W.C. (eds.) Handbook of Coding Theory, pp. 177–294. Elsevier, Amsterdam (1998).

  11. Rains E.M.: Shadow bounds for self-dual-codes. IEEE Trans. Inf. Theory 44, 134–139 (1998).

  12. Simonis J.: MacWilliams identities and coordinate partitions. Linear Algebra Appl. 216, 81–91 (1995).

  13. The GAP Group: GAP—Groups, Algorithms, and Programming, Version 4.7.5 (2014). http://www.gap-system.org.

  14. Yorgov V., Yankov N.: On the extremal binary codes of lengths 36 and 38 with an automorphism of order 5. In: Proceedings of the Fifth International Workshop ACCT, Sozopol, Bulgaria, pp. 307–312 (1996).

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Informatics, Veliko Tarnovo University, Veliko Tarnovo, Bulgaria

    Stefka Bouyuklieva

  2. Otto-von-Guericke Universität, Magdeburg, Germany

    Wolfgang Willems

  3. Faculty of Mathematics and Informatics, Shumen University, Shumen, Bulgaria

    Nikolay Yankov

Authors
  1. Stefka Bouyuklieva
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Wolfgang Willems
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Nikolay Yankov
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Stefka Bouyuklieva.

Additional information

Communicated by J. D. Key.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bouyuklieva, S., Willems, W. & Yankov, N. On the automorphisms of order 15 for a binary self-dual \([96, 48, 20]\) code. Des. Codes Cryptogr. 79, 171–182 (2016). https://doi.org/10.1007/s10623-015-0043-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-015-0043-0

Keywords

Mathematics Subject Classification