Skip to main content
SpringerLink
Log in

New Extremal Type I Codes of Lengths 40, 42, and 44

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

Abstract

It is known that it is possible to construct a generator matrix for a self-dual code of length 2n+2 from a generator matrix of a self-dual code of length 2n. With the aid of a computer, we construct new extremal Type I codes of lengths 40, 42, and 44 from extremal self-dual codes of lengths 38, 40, and 42 respectively. Among them are seven extremal Type I codes of length 44 whose weight enumerator is 1+224y 8+872y 10+·. A Type I code of length 44 with this weight enumerator was not known to exist previously.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. A. Brualdi and V. S. Pless, Weight enumerators of self-dual codes, IEEE Trans. Inform. Theory, Vol. 37 (1991) pp. 1222–1225.

    Google Scholar 

  2. I. Boukliev and S. Buyuklieva, Some new extremal self-dual codes with lengths 44, 50, 54, and 58, IEEE Trans. Inform. Theory, Vol. 44 (1998) pp. 809–812.

    Google Scholar 

  3. S. Buyuklieva, A method for constructing self-dual codes with an automorphism of order 2, IEEE Trans. Inform. Theory, Vol. 46 (2000) pp. 496–504.

    Google Scholar 

  4. S. Buyuklieva, New extremal self-dual codes of lengths 42 and 44, IEEE Trans. Inform. Theory, Vol. (43) (1997) pp. 1607–1612.

    Google Scholar 

  5. S. Buyuklieva, On the binary self-dual codes with an automorphism of order 2, Des. Codes Cryptogr., Vol. 12 (1997) pp. 39–48.

    Google Scholar 

  6. S. Buyuklieva and V. Yorgov, Singly even self-dual codes of length 40, Des. Codes Cryptogr., Vol. 9 (1996) pp. 131–141.

    Google Scholar 

  7. F. C. Bussemaker and V. D. Tonchev, New extremal doubly-even codes of length 40 derived from Hadamard matrices of order 20, Discrete Math., Vol. 82 (1990) pp. 317–321.

    Google Scholar 

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

    Google Scholar 

  9. D. B. Dalan, Type I neighbors of extremal Type II codes of length 40 derived from Hadamard matrices, submitted.

  10. S. T. Dougherty, T. A. Gulliver and M. Harada, Extremal binary self-dual codes, IEEE Trans. Inform. Theory, Vol. 43 (1997) pp. 2036–2047.

    Google Scholar 

  11. M. Harada, Existence of new extremal doubly-even codes and extremal singly-even codes, Des. Codes Cryptogr., Vol. (8) (1996) pp. 1–12.

    Google Scholar 

  12. M. Harada, Existence of new extremal doubly-even and self-dual codes, Des. Codes Cryptogr., Vol. 8 (1996) pp. 273–283.

    Google Scholar 

  13. M. Harada, Weighing matrices and self-dual codes, Ars Combin., Vol. 47 (1997) pp. 65–73.

    Google Scholar 

  14. M. Harada, The existence of a self-dual [70, 35, 12] code and formally self-dual codes, Finite Fields Appl., Vol. 3 (1997) pp. 131–139.

    Google Scholar 

  15. M. Harada, T. A. Gulliver and H. Kaneta, Classification of extremal double circulant self-dual codes of length up to 62, Discrete Math., Vol. 188 (1998) pp. 127–136.

    Google Scholar 

  16. M. Harada and H. Kimura, On extremal self-dual codes, Math. J. Okayama Univ., Vol. 37 (1995) pp. 1–14.

    Google Scholar 

  17. M. Harada and V. D. Tonchev, Self-dual singly-even codes and Hadamard matrices, In Lecture Notes in Computer Science, Proc. AAECC 11. no. 948, Springer-Verlag, New York (1995) pp. 279–284.

    Google Scholar 

  18. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-correcting Codes, North-Holland, New York (1977).

    Google Scholar 

  19. C. L. Mallows and N. J. A. Sloane, An upper bound for self-dual codes, Inform. and Control, Vol. 22 (1973) pp. 188–200.

    Google Scholar 

  20. E. Rains, Shadow bounds for self-dual codes, IEEE Trans. Inform. Theory, Vol. 44 (1998) pp. 134–139.

    Google Scholar 

  21. R. Ruseva and V. Y. Yorgov, Two extremal codes of length 42 and 44, Probl. Pered. Inform., Vol. 29 (1993) pp. 99–103 (in Russian).

    Google Scholar 

  22. E. Rains and N. J. A Sloane, Self-dual codes, In (V. S. Pless and W. C. Huffman, eds.), Handbook of Coding Theory, Elsevier, Amsterdam (1998).

    Google Scholar 

  23. E. Spence, Symmetric (41, 16, 6)-designs with a nontrivial automorphism of odd order, J. Comb. Designs, Vol. 1 (1993) pp. 193–211.

    Google Scholar 

  24. E. Spence and V. D. Tonchev, Extremal self-dual codes from symmetric designs, Discrete Math, Vol. 110 (1992) pp. 265–268.

    Google Scholar 

  25. H.-P. Tsai, Existence of certain extremal self-dual codes, IEEE Trans. Inform. Theory, Vol. 38 (1992) pp. 501–504.

    Google Scholar 

  26. V. Y. Yorgov, The extremal codes of length 42 with automorphism of order 7, Discrete Math., Vol. 190 (1998) pp. 201–213.

    Google Scholar 

  27. V. Y. Yorgov, New extremal singly-even self-dual codes of length 44, In Proceedings of 6th Joint Swedish-Russian Int, Workshop on Information Theory, Mölle, Sweden (1993) pp. 372–375.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Graduate School of Mathematics 33, Kyushu University, Fukuoka, 812–8581, Japan

    Daniel B. Dalan

Authors
  1. Daniel B. Dalan
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dalan, D.B. New Extremal Type I Codes of Lengths 40, 42, and 44. Designs, Codes and Cryptography 30, 151–157 (2003). https://doi.org/10.1023/A:1025476619824

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1025476619824

Search

Navigation