Computing Weierstrass semigroups and the Feng-Rao distance from singular plane models
The authors present an algorithm to compute the Weierstrass semigroup at a point \(P\) and bases for the spaces \(L(mP)\) provided \(P\) is the only branch at infinity of a (possibly singular) plane model of a curve. First, assuming that the characteristic of the field does not divide the degree of the curve, a semigroup at \(P\) on the plane model is found by using the theory of approximate \(d\)-th roots of \textit{S. S. Abhyankar} [``Lectures on expansion techniques in algebraic geometry'', Lectures on Mathematics and Physics. Tata Institute of Fundamental Research. 57 (Bombay 1977; Zbl 0818.14001)]. This semigroup is then augmented to give the semigroup of \(P\) on the normalization of the curve by using an integral basis algorithm [cf. \textit{M. van Hoeij}, J. Symb. Comput. 18,No. 4, 353-363; (1994; Zbl 0834.68059)]. Another algorithm for computing Weierstrass semigroups from a plane model is the classical Brill-Noether algorithm [cf. \textit{D. Le Brigand} and \textit{J.-J. Risler}, Bull. Soc. Math. Fr. 116, No. 2, 231-253 (1988; Zbl 0721.14015)]. The authors then apply their results to algebraic geometry codes. In particular, they show how to compute the order bound [cf. \textit{T. Høholdt, J. H. van Lint} and \textit{R. Pellikaan}, ``Algebraic geometry codes'' in: Handbook of coding theory, 871-961 (1998; Zbl 0922.94015)] for one-point algebraic geometry codes by using the Apéry sequence of the semigroup at that point. [This bound was called the Feng-Rao bound by \textit{C. Kirfel} and \textit{R. Pellikaan} [IEEE Trans. Inf. Theory 41, No. 6, part 1, 1720-1732 (1995; Zbl 0860.94031)].
scientific article; zbMATH DE number 1421408