Symbolic Hamburger-Noether expressions of plane curves and construction of AG codes

TL;DR

This paper introduces algorithms based on symbolic Hamburger-Noether expressions for computing bases of Riemann-Roch spaces and Weierstrass semigroups on algebraic curves, aiding in the construction and decoding of algebraic geometry codes.

Contribution

It develops new symbolic algorithms for algebraic geometry code construction and decoding, combining Hamburger-Noether expansions with Brill-Noether theory.

Findings

Algorithms effectively compute bases for R(G) spaces.

Methods determine Weierstrass semigroups at singular points.

Applicable to the construction and decoding of AG codes.

Abstract

We present an algorithm to compute bases for the spaces L(G), provided G is a rational divisor over a non-singular absolutely irreducible algebraic curve, and also another algorithm to compute the Weierstrass semigroup at P together with functions for each value in this semigroup, provided P is a rational branch of a singular plane model for the curve. The method is founded on the Brill-Noether algorithm by combining in a suitable way the theory of Hamburger-Noether expansions and the imposition of virtual passing conditions. Such algorithms are given in terms of symbolic computation by introducing the notion of symbolic Hamburger-Noether expressions. Everything can be applied to the effective construction of Algebraic Geometry codes and also in the decoding problem of such codes, including the case of the Feng and Rao scheme for one-point codes.