Skip to main content
SpringerLink
Log in

A new subclass of cyclic Goppa codes

  • Coding Theory
  • Published:
Problems of Information Transmission Aims and scope Submit manuscript

Abstract

We propose a subclass of cyclic Goppa codes given by separable self-reciprocal Goppa polynomials of degree two. We prove that this subclass contains all reversible cyclic codes of length n, n | (q m ± 1), with a generator polynomial g(x), g(α ±i) = 0, i = 0, 1, α n = 1, α ∈ GF(q 2m).

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. Goppa, V.D., A New Class of Linear Correcting Codes, Probl. Peredachi Inf., 1970, vol. 6, no. 3, pp. 24–30 [Probl. Inf. Trans. (Engl. Transl.), 1970, vol. 6, no. 3, pp. 207–212].

    MATH  MathSciNet  Google Scholar 

  2. MacWilliams, F.J. and Sloane, N.J.A., The Theory of Error-Correcting Codes, Amsterdam: North-Holland, 1977. Translated under the title Teoriya kodov, ispravlyayushchikh oshibki, Moscow: Svyaz’, 1979.

    MATH  Google Scholar 

  3. Berlecamp, E.R. and Moreno, O., Extended Double-Error-Gorrecting Binary Goppa Codes Are Cyclic, IEEE Trans. Inform. Theory, 1973, vol. 19, no. 6, pp. 817–818.

    Article  Google Scholar 

  4. Tzeng, K.K. and Yu, C.Y., Characterization Theorems for Extending Goppa Codes to Cyclic Codes, IEEE Trans. Inform. Theory, 1979, vol. 25, no. 2, pp. 246–250.

    Article  MATH  MathSciNet  Google Scholar 

  5. Tzeng, K.K. and Zimmermann, K., On Extending Goppa Codes to Cyclic Codes, IEEE Trans. Inform. Theory, 1975, vol. 21, no. 6, pp. 712–716.

    Article  MATH  MathSciNet  Google Scholar 

  6. Moreno, O., Symmetries of Binary Goppa Codes, IEEE Trans. Inform. Theory, 1979, vol. 25, no. 5, pp. 609–612.

    Article  MATH  MathSciNet  Google Scholar 

  7. Vishnevetskii, A.L., On Cyclicity of Extended Goppa Codes, Probl. Peredachi Inf., 1982, vol. 18, no. 3, pp. 14–18 [Probl. Inf. Trans. (Engl. Transl.), 1982, vol. 18, no. 3, pp. 171–175].

    MathSciNet  Google Scholar 

  8. Stichtenoth, H., Which Extended Goppa Codes are Cyclic?, J. Combin. Theory, Ser. A, 1989, vol. 51, no. 2, pp. 205–220.

    Article  MATH  MathSciNet  Google Scholar 

  9. Berger, T.P., Goppa and Related Codes Invariant Under a Prescribed Permutation, IEEE Trans. Inform. Theory, 2000, vol. 46, no. 7, pp. 2628–2633.

    Article  MATH  MathSciNet  Google Scholar 

  10. Berger, T.P., New Classes of Cyclic Extended Goppa Codes, IEEE Trans. Inform. Theory, 1999, vol. 45, no. 4, pp. 1264–1266.

    Article  MATH  MathSciNet  Google Scholar 

  11. Berger, T.P., On the Cyclicity of Goppa Codes, Parity-Check Subcodes of Goppa Codes, and Extended Goppa Codes, Finite Fields Appl., 2000, vol. 6, no. 3, pp. 255–281.

    Article  MATH  MathSciNet  Google Scholar 

  12. Bezzateev, S.V. and Shekhunova, N.A., Subclass of Binary Goppa Codes with Minimal Distance Equal to the Design Distance, IEEE Trans. Inform. Theory, 1995, vol. 41, no. 2, pp. 554–555.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. St. Petersburg State University of Aerospace Instrumentation, St. Petersburg, Russia

    S. V. Bezzateev & N. A. Shekhunova

Authors
  1. S. V. Bezzateev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. A. Shekhunova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. V. Bezzateev.

Additional information

Original Russian Text © S.V. Bezzateev, N.A. Shekhunova, 2013, published in Problemy Peredachi Informatsii, 2013, Vol. 49, No. 4, pp. 57–63.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bezzateev, S.V., Shekhunova, N.A. A new subclass of cyclic Goppa codes. Probl Inf Transm 49, 348–353 (2013). https://doi.org/10.1134/S0032946013040054

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0032946013040054

Keywords

Search

Navigation