Advertisement

Eurocode ’92 pp 135-146 | Cite as

Bound for Trace-Equation and Application to Coding Theory

  • V. Gillot
Part of the International Centre for Mechanical Sciences book series (CISM, volume 339)

Abstract

Using the vector space structure of finite field extension of characteristic two, we transform a one-variable polynomial into a two-variable one. Applying the Deligne bound about exponential sums [1] to the two-variable polynomial, we improve in several cases the bound on the number of solutions for trace-equation. In this way, we obtain different results in Coding Theory. Using the trace-description of the codewords for cyclic codes introduced by Wolfmann [6], we improve the bound on weights for these codes and we also generalize, in different cases, the results of Moreno and Kumar in [2].

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    P. DELIGNE, “La conjecture de Weil I”, Institut Hautes Etudes Sci. Publ. Math., N° 43, (1974).Google Scholar
  2. [2]
    P. KUMAR AND O. MORENO, “Minimum distance bounds for cyclic codes and Deligne’s theorem”, preprint.Google Scholar
  3. [3]
    P. KUMAR AND O. MORENO, “Prime-phase Sequences with Periodic Correlation Properties Better than Binary Sequences”, IEEE Trans. Inform. Theory, vol. 37, 3, May 1991.CrossRefGoogle Scholar
  4. [4]
    R. LIDL AND H. NIEDERREITER, Finite Fields, Encyclopmdia of Mathematics and its applications 20, Addison-Wesley, Reading (1983).Google Scholar
  5. [5]
    F.J. MACWILLIAMS AND N.J.A. SLOANE, The theory of Error-correcting codes, North-Holland, Amsterdam (1977).zbMATHGoogle Scholar
  6. [6]
    J. WOLFMANN, “ New bounds on cyclic codes from algebraic curves”, Lectures Notes in Computer Science, -388- pp 47–62, (1989).Google Scholar
  7. [7]
    J. WOLFMANN, “Polynomial description of binary linear codes and related properties”, A.A.E.C.C., 2, pp 119–138 (1991).zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Wien 1993

Authors and Affiliations

  • V. Gillot
    • 1
  1. 1.University of Toulon and VarLa GardeFrance

Personalised recommendations