[Paper Review] Symbolic Hamburger-Noether expressions of plane curves and construction of AG codes

Antonio Campillo, José Ignacio Farrán|ArXiv.org|Oct 28, 1999

Coding theory and cryptography11 references3 citations

TL;DR

This paper introduces symbolic Hamburger-Noether expressions to algorithmically compute Riemann-Roch spaces L(G) and Weierstrass semigroups at rational singular points of plane curves, enabling effective construction and decoding of Algebraic Geometry (AG) codes, including Feng-Rao one-point codes, via symbolic computation combining Hamburger-Noether expansions and virtual passing conditions.

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.

Motivation & Objective

• To develop algorithmic methods for computing Riemann-Roch spaces L(G) on non-singular curves when G is a rational divisor.
• To compute the Weierstrass semigroup and associated functions at a rational branch of a singular plane curve model.
• To provide a symbolic framework that integrates Hamburger-Noether expansions with virtual passing conditions for algorithmic effectiveness.
• To support the construction and decoding of Algebraic Geometry codes, particularly one-point codes under the Feng-Rao scheme.
• To enable practical implementation of AG codes using symbolic computation techniques grounded in algebraic geometry.

Proposed method

• Introduce symbolic Hamburger-Noether expressions as a computational tool to represent and manipulate local curve singularities algebraically.
• Combine the Brill-Noether algorithm with Hamburger-Noether expansions to systematically compute bases for L(G) spaces.
• Apply virtual passing conditions to enforce divisor conditions symbolically, ensuring correctness in the computation of linear series.
• Use the symbolic framework to extract generators of the Weierstrass semigroup and corresponding functions for each gap and non-gap.
• Implement the algorithms in a way compatible with symbolic computation systems, ensuring reproducibility and efficiency.
• Leverage the structure of plane curve singularities to reduce the complexity of computing Weierstrass semigroups at rational branches.

Experimental results

Research questions

• RQ1 How can Riemann-Roch spaces L(G) be effectively computed for rational divisors on non-singular algebraic curves using symbolic methods?
• RQ2 What symbolic representation enables the algorithmic computation of Weierstrass semigroups at singular rational branches of plane curves?
• RQ3 How can Hamburger-Noether expansions be integrated with virtual passing conditions to compute linear series algorithmically?
• RQ4 In what way can symbolic Hamburger-Noether expressions support the construction of one-point AG codes under the Feng-Rao scheme?
• RQ5 What is the computational framework that allows both construction and decoding of AG codes via symbolic algebraic geometry?

Key findings

• The authors present a complete algorithmic framework for computing bases of L(G) spaces using symbolic Hamburger-Noether expressions.
• The method successfully computes the Weierstrass semigroup and associated functions at a rational singular point of a plane curve.
• The integration of Hamburger-Noether expansions with virtual passing conditions enables precise and systematic computation of linear series.
• The approach is effective for constructing one-point AG codes, including those under the Feng-Rao scheme, via symbolic computation.
• The symbolic framework ensures that all computations are algorithmic, reproducible, and suitable for implementation in computer algebra systems.
• The method provides a constructive solution to the decoding problem for AG codes by leveraging the computed semigroup structure and function bases.