Designs, Codes and Cryptography

, Volume 87, Issue 2–3, pp 569–587 | Cite as

Totally decomposed cumulative Goppa codes with improved estimations

  • Sergey BezzateevEmail author
  • Natalia Shekhunova
Part of the following topical collections:
  1. Special Issue: Coding and Cryptography


A class of q-ary totally decomposed cumulative \(\varGamma (L,G^{j})\)-codes with \(L=\{ \alpha \in :G(\alpha )\ne 0 \}\) and \(G^{j}=G^{j}(x),\; 1 \le j\le q\), where G(x) is a polynomial totally decomposed in \(GF(q^{m})\), are considered. The relation between different codes from this class is studied. Improved bounds of the minimum distance and dimension are obtained.


Goppa codes Totally decomposed Goppa codes Wild Goppa codes 

Mathematics Subject Classification

94B05 94B65 



  1. 1.
    Bernstein D.J., Lange T., Peters C.: Wild McEliece incognito. In: Yang B.Y. (ed.) Post-Quantum Cryptography, pp. 244–254. Springer, Berlin (2011).CrossRefGoogle Scholar
  2. 2.
    Bezzateev S., Shekhunova N.: On the subcodes of one class Goppa codes. In: Proc. Intern. Workshop Algebraic and Combinatorial Coding Theory, pp. 143–146 (1988).Google Scholar
  3. 3.
    Bezzateev S., Shekhunova N.: Subclass of binary Goppa codes with minimal distance equal to the design distance. IEEE Trans. Inf. Theory 41(2), 554–555 (1995).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bezzateev S.V., Shekhunova N.A.: Quasi-cyclic Goppa codes. In: Proceedings of 1995 IEEE International Symposium on Information Theory, p. 499 (1995).Google Scholar
  5. 5.
    Bezzateev S., Shekhunova N.: Subclass of non-binary cumulative Goppa codes. In: The 9th International Conference on Finite Fields and Their Applications, p. 7. University College Dublin. (2009).
  6. 6.
    Bezzateev S., Shekhunova N.: Cumulative-separable codes. arXiv:1005.1524v1 (2010).
  7. 7.
    Bezzateev S., Shekhunova N.: Special classes of Goppa codes with improved estimations for parameters. Probl. Peredachi Inf. 46(3), 29–50 (2010).MathSciNetzbMATHGoogle Scholar
  8. 8.
    Couvreur A., Otmani A., Tillich J.P.: New identities relating wild Goppa codes. Finite Fields Appl. 29, 178–197 (2014). Scholar
  9. 9.
    Goppa V.: A new class of linear error correcting codes. Probl. Peredachi Inf. 6(3), 24–30 (1970).MathSciNetzbMATHGoogle Scholar
  10. 10.
    Grassl M.: Bounds on the minimum distance of linear codes and quantum codes. (2007). Accessed 16 June 2018
  11. 11.
    MacWilliams F., Sloane N.: The Theory of Error-Correcting Codes. North-Holland Publishing Co, Amsterdam. (1977).
  12. 12.
    Sugiyama Y., Kasahara M., Hirasawa S., Namekawa T.: Further results on Goppa codes and their applications to constructing efficient binary codes. IEEE Trans. Inf. Theory 22(5), 518–526 (1976). Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Saint Petersburg University of Aerospace InstrumentationSaint PetersburgRussia

Personalised recommendations