Designs, Codes and Cryptography

, Volume 36, Issue 3, pp 317–325 | Cite as

Proof of Conjectures on the True Dimension of Some Binary Goppa Codes

Article

Abstract

There is a classical lower bound on the dimension of a binary Goppa code. We survey results on some specific codes whose dimension exceeds this bound, and prove two conjectures on the true dimension of two classes of such codes.

Keywords

Goppa codes trace operator redundancy equation parameters of Goppa codes 

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References

  1. 1.
    S. V. Bezzateev, E. T. Mironchikov and N. A. Shekhunova, One subclass of binary Goppa codes, Proc. XI Simp. po Probl. Izbit. v Inform. Syst., (1986) pp. 140–141.Google Scholar
  2. 2.
    S. V. Bezzateev and N. A. Shekhunova, On the subcodes of one class Goppa Codes, Proc. Intern. Workshop Algebraic and Combinatorial Coding Theory ACCT-1 (1988) pp. 143–146.Google Scholar
  3. 3.
    Bezzateev, S. V., Mironchikov, E. T., Shekhunova, N. A. 1989A subclass of binary Goppa codeProblemy Peredachi Informatsii2598102Google Scholar
  4. 4.
    Bezzateev, S. V., Shekhunova, N. A. 1995Subclass of binary Goppa codes with minimal distance equal to the design distanceIEEE Transactions on Information Theory41554555CrossRefGoogle Scholar
  5. 5.
    Bezzateev, S. V., Shekhunova, N. A. 1998A subclass of binary Goppa codes with improved estimation of the code dimensionDesigns Codes and Cryptography142338CrossRefGoogle Scholar
  6. 6.
    Delsarte, P. 1975On subfield subcodes of modified Reed-Solomon codesIEEE Transactions on Information TheoryIT-21575576CrossRefGoogle Scholar
  7. 7.
    J. K. Gibson, Equivalent Goppa codes and trapdoors to McEliece’s public key cryptosystem, In Advances in Cryptology—Eurocrypt’91, LNCS No. 547, Springer-Verlag (1991) pp. 517–521.Google Scholar
  8. 8.
    Goppa, V. D. 1970A new class of linear error correcting codesProblemy Peredachi Informatsii62430Google Scholar
  9. 9.
    Handbook of Coding Theory, Vol. 1, V. S. Pless and W. C. Huffman (ed.), NorthHolland (1998).Google Scholar
  10. 10.
    Jensen, J. M. 1995Subgroup subcodesIEEE Transactions on Information Theory41781785CrossRefGoogle Scholar
  11. 11.
    Loeloeian, M., Conan, J. 1987A transform approach to Goppa codesIEEE Transactions on Information TheoryIT-33105115CrossRefGoogle Scholar
  12. 12.
    F. J. Mac Williams and N. J. A. Sloane, The Theory of Error Correcting Codes, North Holland (1983).Google Scholar
  13. 13.
    A. M. Roseiro, The trace operator and generalized Goppa codes, Ph.D. Dissert., Dept. of Elect. Eng., Michigan State Univ., East Lansing, MI 48823 (1989).Google Scholar
  14. 14.
    Roseiro, A. M., Hall, J. I., Hadney, J. E., Siegel, M. 1992The trace operator and redundancy of Goppa codesIEEE Transactions on Information Theory3811301133CrossRefGoogle Scholar
  15. 15.
    Stichtenoth, H. 1990On the dimension of subfield subcodesIEEE Transactions on Information Theory369093CrossRefGoogle Scholar
  16. 16.
    Vlugt, M. 1990The true dimension of certain binary Goppa codesIEEE Transactions on Information Theory36397398CrossRefGoogle Scholar
  17. 17.
    Vlugt, M. 1991On the dimension of trace codesIEEE Transactions on Information Theory37196199CrossRefGoogle Scholar
  18. 18.
    Véron, P. 1998Goppa codes and trace operatorIEEE Transactions on Information Theory44290295CrossRefGoogle Scholar
  19. 19.
    Véron, P. 2001True dimension of some binary quadratic trace Goppa codesDesigns Codes and Cryptography248197CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Groupe de Recherche en Informatique et Mathématiques (GRIM)Université de Toulon-VarCedexFrance

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