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New bounds on cyclic codes from algebraic curves

  • J. Wolfmann
Section I Coding And Algebraic Geometry
Part of the Lecture Notes in Computer Science book series (LNCS, volume 388)

Abstract

Starting from a deep link between the words of cyclic codes and plane algebraic curves over finite fields we use bounds on the number of rational points of these curves to obtain general bounds for the weights of cyclic codes.

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References

  1. [1]
    Assmus Jr, E.F.,Mattson Jr, H.F.,Coding and combinatorics,SIAM Rewiew 16(1974) 349–388CrossRefGoogle Scholar
  2. [2]
    Baumert, L.D., McEliece, R.J.,Weights of irreducible cyclic codes, Information and control 20 (1972), 158–175.CrossRefGoogle Scholar
  3. [3]
    Fulton, W., Algebraic Curves. Lecture Notes, Benjamin, Reading, (1969).Google Scholar
  4. [4]
    Hartshorne, R., Algebraic Geometry, Graduate texts in Math. no 52, Springer, New-York (1977).Google Scholar
  5. [5]
    Lachaud, G., Exponential sums and the Carlitz-Uchiyama bound. (In the same proceedings).Google Scholar
  6. [6]
    Lachaud, G., Wolfmann, J.,Sommes de Kloosterman, courbes elliptiques et codes cycliques en caractéristique 2, C. R. Acad. Sci. Paris (I), 305 (1987), p.881–883.Google Scholar
  7. [7]
    Lachaud, G., Wolfmann, J.,The weights of the orthogonals of the extended quadratic binary Goppa codes.(submitted for publication)Google Scholar
  8. [8]
    Lidl, R., Niederreiter, H., Finite fields, Encyclopedia of Mathematics and its applications, 20, Addison-Wesley, Reading (1983).Google Scholar
  9. [9]
    McEliece, R.J., Irreducible cyclic codes and Gauss sums in "Combinatorics" (M. Hall,Jr. and J.H. van Lint,Eds.),pp 185–202,Reidel, Dordrecht-Boston (1975)Google Scholar
  10. [10]
    McEliece, R.J.,Finite fields for computer scientists and engineers, Kluwer, (1986).Google Scholar
  11. [11]
    McWilliams, F.J., Sloane, N.J.A.,The theory of Error-correcting codes, North-Holland, Amsterdam, (1977).Google Scholar
  12. [12]
    Niederreiter, H, Weights of Cyclic Codes, Information and Control,34 (1977), p.130–140.CrossRefGoogle Scholar
  13. [13]
    Remijn, J.C.C.M., Tiersma, H.J., A duality theorem for the weight distribution of some cyclic codes IEEE Trans. Info. Theory 34 (1988) p.1348–1351.CrossRefGoogle Scholar
  14. [14]
    Serre, J.P., Sur le nombre des points rationnels d'une courbe algébrique sur un corps fini. C. R. Acad. Sci. Paris (I), 296 (1983), p.397–402.Google Scholar
  15. [15]
    Serre,J.P., Nombre de points des courbes algébriques sur F q. Seminaire de Theorie des Nombre dee Bordeaux, exposé no.22 (1983)Google Scholar
  16. [16]
    Stichtenoth, H.,Uber die Automorphismengruppe eines algebraischen Functionenkorpers von Primzahlcharacteristik.teil II: Ein spezieller Typ von Funktionenkorpern. Arch.Math. 24 (1973) p 615–631.CrossRefGoogle Scholar
  17. [17]
    Weil, A., Variétés abéliennes et courbes algébriques, Hermann, Paris,(1948).Google Scholar
  18. [18]
    Wolfmann, J.,Codes projectifs a deux ou trois poids associés aux hyperquadriques d'une géometrie finie, Discrete math. 13 (1975), p.185–211.CrossRefGoogle Scholar
  19. [19]
    Wolfmann, J., The weights of the dual code of the Melas code over GF(3). Discrete math. 74 (1989) p.327–329.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • J. Wolfmann
    • 1
  1. 1.G.E.C.T. Université de ToulonLa GardeFrance

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