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Designs, Codes and Cryptography

, Volume 82, Issue 1–2, pp 69–76 | Cite as

Lower bound of covering radius of binary irreducible Goppa codes

  • Sergey BezzateevEmail author
  • Natalia Shekhunova
Article

Abstract

The lower bound of covering radius of binary irreducible Goppa codes is obtained.

Keywords

Goppa codes Covering radius Lower bound 

Mathematics Subject Classification

94B65 

Notes

Acknowledgments

The research leading to these results has received funding from the Ministry of Education and Science of the Russian Federation according to the project part of the state funding Assignment No. 2.2716.2014/K, July 17th, 2014.

References

  1. 1.
    Bezzateev S.V., Shekhunova N.A.: Class of binary generalized Goppa codes perfect in weighted Hamming metric. In: Proceedings of WCC-2011, Paris, pp. 233–242 (2011)Google Scholar
  2. 2.
    Bezzateev, S., Shekhunova, N.: Class of generalized Goppa codes perfect in weighted Hamming metric. Des. Codes Cryptogr. 66(1–3), 391–399 (2013)Google Scholar
  3. 3.
    Carlitz L.: The arithmetic of polynomials in a Galois field. Am. J. Math. 54, 39–50 (1932)Google Scholar
  4. 4.
    Cohen G., Frankl P.: Good coverings of Hamming spaces with spheres. Discret. Math. 56, 125–131 (1985)Google Scholar
  5. 5.
    Cohen G., Honkala I., Litsyn S., Lobstein A.: Covering Codes. North-Holland, Amsterdam (1997)Google Scholar
  6. 6.
    Cohen G.D., Karpovsky M.G., Mattson Jr. H.F., Schatz J.R.: Covering radius survey and recent results. IEEE Trans. Inf. Theory 31, 328–343 (1985)Google Scholar
  7. 7.
    Cohen G.D., Litsyn S.N., Lobstein A.C., Mattson Jr. H.F.: Covering radius 1985–1994. Appl. Algebra Eng. Commun. Comput. 8, 173–239 (1997)Google Scholar
  8. 8.
    Cohen G.D., Lobstein A.C., Sloane N.J.A.: Further results of the covering radius of codes. IEEE Trans. Inf. Theory 32, 680–694 (1986)Google Scholar
  9. 9.
    Feng G.L., Tzeng K.K.: On Quasi-perfect property of double-error-correcting Goppa codes and their complete decoding. Inf. Control 61, 132–146 (1984)Google Scholar
  10. 10.
    Goppa V.D.: A new class of linear error correcting codes. Probl. Inf. Trans. 6(3), 24–30 (1970)Google Scholar
  11. 11.
    Grassl M., Tomlinson M., Tjhai C.J., Jibril M., Ahmed M.Z.: Results on the covering radius of some best known binary codes, University of Plymouth (2011) (unpublished manuscript)Google Scholar
  12. 12.
    Lang S.: Algebra, Reading Mass (3rd edn.), Addison-Wesley, Boston (1993)Google Scholar
  13. 13.
    Levy-dit-Vehel F., Litsyn S.: Parameters of Goppa codes revisited. IEEE Trans. Inf. Theory, 1811–1819 (1997)Google Scholar
  14. 14.
    MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977)Google Scholar
  15. 15.
    Moreno O.: Goppa Codes Related Quasi-Perfect Double-Error-Correcting Codes. In: IEEE International Symposium Information Theory “Abstracts of Papers”, Santa Monica (1981)Google Scholar
  16. 16.
    Moreno C.J., Moreno O.: Exponential sums and Goppa codes I. Proc. Am. Math. Soc. 111(2), 523–531 (1991)Google Scholar
  17. 17.
    Tietavainen A.: Codes and Character Sums: Coding Theory and Application. Lecture Notes in Computer Science, vol. 388, pp. 3–12. Springer–Verlag, Berlin (1989)Google Scholar
  18. 18.
    Vladuts S.G., Skorobogatov A.N.: Covering radius for long BCH codes, problemy peredachi inlormatsii, vol. 25, 38–45. Translated in: Probl. Inf. Trans., 25(1), 28–34 (1989)Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Saint Petersburg State University of Aerospace InstrumentationSaint PetersburgRussia

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