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Codes and Elliptic Curves

  • Gerard van der Geer
Part of the Progress in Mathematics book series (PM, volume 94)

Abstract

In this paper we discuss recent results ([1], [2], [3], [4], [5], [9]) on codes and algebraic curves. We are not concerned with algebraic geometric Goppa codes but rather with another link between coding theory and algebraic geometry. In our case the codes correspond to families of algebraic curves over a finite field, whereas Goppa codes come from a fixed algebraic curve. We use algebraic geometry to determine the weight distributions of certain codes but also we show that results from coding theory may be used to obtain results about the variation of the number of points in families of algebraic curves over a finite field. We do not assume that the reader has a knowledge of coding theory.

Keywords

Elliptic Curve Finite Field Elliptic Curf Algebraic Curf Cyclic 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.

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References

  1. [1]
    van der Geer, G., Schoof, R., van der Vlugt, M., Weight Formulas for Ternary Melas Codes, Preprint 1990.Google Scholar
  2. [2]
    van der Geer, G., van der Vlugt, M., Ariin-Schreier Curves and Codes, Preprint 1989, Journal of Algebra (to appear).Google Scholar
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    van der Geer, G., van der Vlugt, M., Reed-Muller Codes and Supersingular Curves. I., Preprint 1990.Google Scholar
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    van der Geer, G., van der Vlugt, M., Families of algebraic curves and codes, In preparation.Google Scholar
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    Lachaud, G., Wolfmann, J., Sommes de Kloosterman, courbes elliptiques et codes cycliques en charactéristique 2, Comptes Rendus Acad. Sci. Paris 305 (1987), 881–883.MathSciNetMATHGoogle Scholar
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    van Lint, J. H., “Introduction to Coding Theory,” Grad. Texts in Math., Springer Verlag, 1982.Google Scholar
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    van Lint, J. H., van der Geer, G., “Introduction to Coding Theory and Algebraic Geometry,” Birkhäuser, 1988.Google Scholar
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    MacWilliams, F. J., Sloane, N. J. A., “The Theory of Error Correcting Codes,” North Holland, Amsterdam, 1983.Google Scholar
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    Schoof, R., van der Vlugt, M., Hecke operators and the weight distribution of certain codes, Preprint 1989, Journal of Comb. Theory (to appear).Google Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Gerard van der Geer
    • 1
  1. 1.Faculteit Wiskunde en InformaticaUniversiteit van AmsterdamAmsterdamThe Netherlands

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