Designs, Codes and Cryptography

, Volume 79, Issue 1, pp 171–182 | Cite as

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

  • Stefka Bouyuklieva
  • Wolfgang Willems
  • Nikolay Yankov


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).


Self-dual codes Doubly-even codes Automorphisms 

Mathematics Subject Classification

94B05 20B25 


  1. 1.
    Assmus E.F., Mattson H.F.: New 5-designs. J. Comb. Theory 6, 122–151 (1969).Google Scholar
  2. 2.
    Bouyukliev I.: What is Q-extension? Serdica J. Comput. 1, 115–130 (2007).Google Scholar
  3. 3.
    de la Cruz J., Willems W.: On extremal self-dual codes of length 96. IEEE Trans. Inf. Theory 57, 6820–6823 (2011).Google Scholar
  4. 4.
    Grassl M.: Bounds on the minimum distance of linear codes and quantum codes.
  5. 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).Google Scholar
  6. 6.
    Huffman W.C.: Decomposing and shortening codes using automorphisms. IEEE Trans. Inf. Theory 32, 833–836 (1986).Google Scholar
  7. 7.
    Huffman W.C., Pless V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003).Google Scholar
  8. 8.
    Kéri G.: Tables for \(n\)-arcs and lists for complete \(n\)-arcs in \(PG(r, q)\).
  9. 9.
    Ling S., Solé P.: On the algebraic structure of quasi-cyclic codes I: Finite fields. IEEE Trans. Inf. Theory 47, 2751–2760 (2001).Google Scholar
  10. 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).Google Scholar
  11. 11.
    Rains E.M.: Shadow bounds for self-dual-codes. IEEE Trans. Inf. Theory 44, 134–139 (1998).Google Scholar
  12. 12.
    Simonis J.: MacWilliams identities and coordinate partitions. Linear Algebra Appl. 216, 81–91 (1995).Google Scholar
  13. 13.
    The GAP Group: GAP—Groups, Algorithms, and Programming, Version 4.7.5 (2014).
  14. 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).Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Stefka Bouyuklieva
    • 1
  • Wolfgang Willems
    • 2
  • Nikolay Yankov
    • 3
  1. 1.Faculty of Mathematics and InformaticsVeliko Tarnovo UniversityVeliko TarnovoBulgaria
  2. 2.Otto-von-Guericke UniversitätMagdeburgGermany
  3. 3.Faculty of Mathematics and InformaticsShumen UniversityShumenBulgaria

Personalised recommendations