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Zbl 0721.14015
Le Brigand, D.; Risler, J.J.
Algorithme de Brill-Noether et codes de Goppa. (Brill-Noether algorithm and Goppa codes). (French)
[J] Bull. Soc. Math. Fr. 116, No.2, 231-253 (1988). ISSN 0037-9484

Summary: Let C be a projective plane curve, defined on a field k, with ordinary singular points; the classical Brill-Noether algorithm gives a construction of a basis of the k-vector space ${\cal L}(D)$, D being an effective, k-rational, divisor of C. \par In this paper, we present a generalisation of the Brill-Noether algorithm to a projective plane curve having non ordinary singularities. In the frame work of code theory, that generalisation gives a construction of ``good'' codes associated with curves having many rational points over a given finite field. We give, as an example, the construction of a code, the parameters of which are better than Varshamov-Gilbert bound; this code is associated with a curve having a non ordinary singular point and the maximum number of rational points over ${\bbfF}\sb{16}$ within the Weil bound.

MSC 2000:: *14H20 Singularities, local rings
94B05 General theory of linear codes
14C20 Divisors, linear systems, invertible sheaves

Keywords: rational divisor; Goppa codes; Brill-Noether algorithm; finite field

Cited in: Zbl 0972.14025 Zbl 0876.94047 Zbl 0853.94024

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