On Weierstrass semigroups and algebraic geometry one-point codes

Abstract

We present two different algorithms to compute the Weierstrass semigroup at a point P together with functions for each value in this semigroup from a plane model of the curve. The first one works in a quite general situation and it is founded on the Brill-Noether algorithm. The second method works in the case of P being the only point at infinity of the plane model, what is very usual in practice, and it is based on the Abhyankar-Moh theorem, the theory of approximate roots and an integral basis for the affine algebra of the curve. This last way is simpler and has an additional advantage: one can easily compute the Feng-Rao distances for the corresponding array of one-point algebraic geometry codes, this thing be done by means of the Apéry set of the Weierstrass semigroup. Everything can be applied to the problem of decoding such codes by using the majority scheme of Feng and Rao.