Computation in algebraic function fields for effective construction of algebraic-geometric codes

  • Gaétan Haché
Submitted Contributions
Part of the Lecture Notes in Computer Science book series (LNCS, volume 948)


We present a method for effective construction of algebraic-geometric codes based on the Brill-Noether algorithm. This paper is based on a paper by Le Brigand and Risler [8], but the presentation uses only the theory of algebraic function fields of one variable.


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  1. 1.
    M. Bronstein, M. Hassner, A. Vasquez and C.J. Williamson, Algebraic algorithms for the construction of error correction codes on algebraic curves, Proceedings of IEEE International Symposium on Information Theory, June 1991.Google Scholar
  2. 2.
    W. Fulton, Algebraic curves: An introduction to algebraic geometry, W.A. Benjamin, Inc, New-York, Amsterdam, 1969.Google Scholar
  3. 3.
    V.D. Goppa, Codes associated with divisors, Probl. Peredach. infor., 13(1):33–39, 1977.Google Scholar
  4. 4.
    D. Gorenstein, An arithmetic theory of adjoint plane curves, Trans. Amer. Math. Soc. 72 (1952), 414–436.Google Scholar
  5. 5.
    G. Haché and D. Le Brigand, Effective Construction of Algebraic Geometry Codes, Technical Report 2267, INRIA, May 1994.Google Scholar
  6. 6.
    H. Hironaka, On the arithmetic genera and the effective genera of algebraic curves, Memoirs of the College of Sciences of Kyoto, Series A, 30, Math. 2 (1957),177–195.Google Scholar
  7. 7.
    D. Lazard, Solving zero-dimensional algebraic systems, J. Symbolic Cumputation, 13, 1992.Google Scholar
  8. 8.
    D. Le Brigand and J.J. Risler, Algorithme de Brill-Noether et codes de Goppa, Bull. Soc. math. France, 116 (1988), 231–253.Google Scholar
  9. 9.
    D. Polemi, M. Hassner, O. Moreno and C.J. Williamson, A computer algebra algorithm for the adjoint divisor, Proceedings of IEEE International Symposium on Information Theory, January 1993.Google Scholar
  10. 10.
    H. Stichtenoth, Algebraic function fields and codes, University Text, Springer-Verlag, 1993.Google Scholar
  11. 11.
    M.Tsfasman and S. Vladut, Algebraic-geometric codes, Kluwer Academic Pub., Math. and its Appl. 58, 1991.Google Scholar
  12. 12.
    A.T. Vasquez, Rational desingularization of a curve defined over a finite field, Number Theory, N. Y. Seminar 1989–1990, Springer-Verlag, 229–250.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Gaétan Haché
    • 1
  1. 1.Projet CODESINRIA-RocquencourtLe Chesnay CedexFrance

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