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Part of the book series: Progress in Mathematics ((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.

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References

  1. van der Geer, G., Schoof, R., van der Vlugt, M., Weight Formulas for Ternary Melas Codes, Preprint 1990.

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  2. van der Geer, G., van der Vlugt, M., Ariin-Schreier Curves and Codes, Preprint 1989, Journal of Algebra (to appear).

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  3. van der Geer, G., van der Vlugt, M., Reed-Muller Codes and Supersingular Curves. I., Preprint 1990.

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  4. van der Geer, G., van der Vlugt, M., Families of algebraic curves and codes, In preparation.

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  5. Lachaud, G., Wolfmann, J., Sommes de Kloosterman, courbes elliptiques et codes cycliques en charactéristique 2, Comptes Rendus Acad. Sci. Paris 305 (1987), 881–883.

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  6. van Lint, J. H., “Introduction to Coding Theory,” Grad. Texts in Math., Springer Verlag, 1982.

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  7. van Lint, J. H., van der Geer, G., “Introduction to Coding Theory and Algebraic Geometry,” Birkhäuser, 1988.

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  8. MacWilliams, F. J., Sloane, N. J. A., “The Theory of Error Correcting Codes,” North Holland, Amsterdam, 1983.

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  9. Schoof, R., van der Vlugt, M., Hecke operators and the weight distribution of certain codes, Preprint 1989, Journal of Comb. Theory (to appear).

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© 1991 Springer Science+Business Media New York

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van der Geer, G. (1991). Codes and Elliptic Curves. In: Mora, T., Traverso, C. (eds) Effective Methods in Algebraic Geometry. Progress in Mathematics, vol 94. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0441-1_10

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  • DOI: https://doi.org/10.1007/978-1-4612-0441-1_10

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6761-4

  • Online ISBN: 978-1-4612-0441-1

  • eBook Packages: Springer Book Archive

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