Skip to main content
Log in

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

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

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.

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

Access this article

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. 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.

  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.

  3. S. V. Bezzateev E. T. Mironchikov N. A. Shekhunova (1989) ArticleTitleA subclass of binary Goppa code Problemy Peredachi Informatsii 25 IssueID3 98–102

    Google Scholar 

  4. S. V. Bezzateev N. A. Shekhunova (1995) ArticleTitleSubclass of binary Goppa codes with minimal distance equal to the design distance IEEE Transactions on Information Theory 41 IssueID2 554–555 Occurrence Handle10.1109/18.370170

    Article  Google Scholar 

  5. S. V. Bezzateev N. A. Shekhunova (1998) ArticleTitleA subclass of binary Goppa codes with improved estimation of the code dimension Designs Codes and Cryptography 14 IssueID1 23–38 Occurrence Handle10.1023/A:1008252303768

    Article  Google Scholar 

  6. P. Delsarte (1975) ArticleTitleOn subfield subcodes of modified Reed-Solomon codes IEEE Transactions on Information Theory IT-21 575–576 Occurrence Handle10.1109/TIT.1975.1055435

    Article  Google Scholar 

  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.

  8. V. D. Goppa (1970) ArticleTitleA new class of linear error correcting codes Problemy Peredachi Informatsii 6 24–30

    Google Scholar 

  9. Handbook of Coding Theory, Vol. 1, V. S. Pless and W. C. Huffman (ed.), NorthHolland (1998).

  10. J. M. Jensen (1995) ArticleTitleSubgroup subcodes IEEE Transactions on Information Theory 41 IssueID3 781–785 Occurrence Handle10.1109/18.382025

    Article  Google Scholar 

  11. M. Loeloeian J. Conan (1987) ArticleTitleA transform approach to Goppa codes IEEE Transactions on Information Theory IT-33 105–115 Occurrence Handle10.1109/TIT.1987.1057276

    Article  Google Scholar 

  12. F. J. Mac Williams and N. J. A. Sloane, The Theory of Error Correcting Codes, North Holland (1983).

  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).

  14. A. M. Roseiro J. I. Hall J. E. Hadney M. Siegel (1992) ArticleTitleThe trace operator and redundancy of Goppa codes IEEE Transactions on Information Theory 38 IssueID3 1130–1133 Occurrence Handle10.1109/18.135654

    Article  Google Scholar 

  15. H. Stichtenoth (1990) ArticleTitleOn the dimension of subfield subcodes IEEE Transactions on Information Theory 36 90–93 Occurrence Handle10.1109/18.50376

    Article  Google Scholar 

  16. M. Vlugt Particlevan der (1990) ArticleTitleThe true dimension of certain binary Goppa codes IEEE Transactions on Information Theory 36 IssueID2 397–398 Occurrence Handle10.1109/18.52487

    Article  Google Scholar 

  17. M. Vlugt Particlevan der (1991) ArticleTitleOn the dimension of trace codes IEEE Transactions on Information Theory 37 IssueID1 196–199 Occurrence Handle10.1109/18.61140

    Article  Google Scholar 

  18. P. Véron (1998) ArticleTitleGoppa codes and trace operator IEEE Transactions on Information Theory 44 IssueID1 290–295 Occurrence Handle10.1109/18.651048

    Article  Google Scholar 

  19. P. Véron (2001) ArticleTitleTrue dimension of some binary quadratic trace Goppa codes Designs Codes and Cryptography 24 IssueID1 81–97 Occurrence Handle10.1023/A:1011281431366

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to P. Véron.

Additional information

Part of this work has been presented at the Sixth International Conference on Finite Fields and Applications, Oaxaca, Mexico, May 2001.

AMS classification: 94B65

Rights and permissions

Reprints and permissions

About this article

Cite this article

Véron, P. Proof of Conjectures on the True Dimension of Some Binary Goppa Codes. Des Codes Crypt 36, 317–325 (2005). https://doi.org/10.1007/s10623-004-1722-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-004-1722-4

Keywords

Navigation