Advertisement

Polynomial-time construction of linear codes with almost equal weights

  • Gilles Lachaud
  • Jacques Stern
Conference paper

Abstract

If C is a binary linear code with d ≤ w(x) ≤ D for every non-zero codeword x in C, we define the disparity of C to be the ratio
$$ r(C) = \frac{D}{d} $$
The single-weight codes are those with r = 1. Any single-weight binary linear code C of dimension k and length n is such that
$$ k \leqslant 1 + {\log_2}(n) $$
thus, there is no infinite family of single-weight codes whose length is linearly bounded with respect to the dimension.

Keywords

Equal Weight Information Transmission Binary Code Discrete Math Linear Code 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Goppa, V.D., Algebraico-geometric codes, Izv. Akad. Nauk S.S.S.R., 46(1982)Google Scholar
  2. [1a]
    Goppa, V.D., Algebraico-geometric codes, Math. U.S.S.R Izvestiya, 21 (1983), 75–91.CrossRefGoogle Scholar
  3. [2]
    Goppa, V.D., Geometry and codes, Kluwer Acad. Pub., Dordrecht, 1988.MATHGoogle Scholar
  4. [3]
    Katsman, G.L., Tsfasman, M.A. and Vlǎdut, S.G., Modular Curves and Codes with polynomial complexity of construction, Problemy Peredachi Informatsii 20(1984), 47–55;MathSciNetGoogle Scholar
  5. [3a]
    Katsman, G.L., Tsfasman, M.A. and Vlǎdut, S.G., Modular Curves and Codes with polynomial complexity of construction, Problems of information Transmission 20 (1984), 35–42.MathSciNetMATHGoogle Scholar
  6. [4]
    Katsman, G.L., Tfasman, M.A. and Vlàdut, S.G., Modular Curves and Codes with a polynomial construction, IEEE Transactions on Information Theory, 30 (1984), 353–355.MATHCrossRefGoogle Scholar
  7. [5]
    Lachaud, G., Les codes géométriques de Goppa, Séminaire Bourbaki 1984/85, exp. 641, Ast¡’erisque 133–134, 189–207.Google Scholar
  8. [6]
    Lachaud, G., Projective Reed-Muller codes, Coding Theory and Applications, Proc. 2nd Int. Coll. Paris 1986, Lect. Notes in Comp. Sci. 311 (1988), 125–129.MathSciNetGoogle Scholar
  9. [7]
    Lachaud, G., The parameters of projective Reed-Muller codes, Discrete Math. 81 (1990), 1–5.MathSciNetCrossRefGoogle Scholar
  10. [8]
    McWilliams, F.J., Sloane, N.J.A., The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1977.Google Scholar
  11. [9]
    Manin, Yu. I., Vlǎdut, S.G., Codes linéaires et Courbes Modulaires, Itogi Nauki i Tekhniki 25(1984), 209–257;Google Scholar
  12. [9a]
    Manin, Yu. I., Vlǎdut, S.G., Codes linéaires et Courbes Modulaires, J. Soviet Math. 30 (1985), 2611–1643.MATHCrossRefGoogle Scholar
  13. [10]
    Tsfasman, M.A., Vlàdut, S.G., Zink, Th., Modular Curves, Shimura Curves and Goppa Codes better than Varshamov-Gilbert bound, Math. Nachr. 109 (1982), 21–28.MathSciNetMATHCrossRefGoogle Scholar
  14. [11]
    Tsfasman, M.A., Vlàdut, S.G., Algebraico-geometric Codes, Kluwer Acad. Pub., Dordrecht (1991).Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1993

Authors and Affiliations

  • Gilles Lachaud
    • 1
  • Jacques Stern
    • 2
  1. 1.Equipe A.T.I., C.I.R.M.Marseille-LuminyFrance
  2. 2.G.R.E.C.C., D.M.I.École Normale SupérieureFrance

Personalised recommendations