Symbolic Hamburger-Noether expressions of plane curves and construction of AG codes
A. Campillo, J. I. Farran
TL;DR
This work addresses the practical computation of spaces $\mathcal{L}(G)$ and Weierstrass semigroups using symbolic Hamburger-Noether expressions on plane curve models. It fuses Hamburger-Noether theory with Brill-Noether methods and Enriques' discharge to impose adjunction conditions, yielding effective bases and explicit pole functions on the curve. The resulting framework enables productive construction and decoding of Algebraic Geometry codes, including Feng-Rao one-point schemes, by providing the necessary semigroup and function data. The authors implement the algorithms in SINGULAR and present a unified symbolic approach to desingularization, adjunction, and coding-theory applications.
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.