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Computing Weierstrass semigroups and the Feng-Rao distance from singular plane models

A. Campillo, J. I. Farran

TL;DR

The paper develops an algorithm to compute the Weierstrass semigroup $Γ_P$ at a point $P$ on a curve with a singular plane model having a single branch at infinity, using Abhyankar–Moh theory and approximate roots, complemented by a triangulation step from the integral closure. This yields not only a basis for spaces $L(mP)$ but also a practical route to compute the Feng–Rao distance for corresponding one-point AG codes, with a general Apéry-system framework that yields $oldsymbol{δ_{FR}(m)}$ for arbitrary semigroups and refinements for symmetric and infinity-related semigroups. The results enable improved decoding of algebraic geometry codes through the Feng–Rao majority scheme and strengthen the bridge between singularity theory and coding theory by providing constructive semigroup descriptions and pole-order functions. Practically, the approach supports efficient code construction, explicit bases for $L(mP)$, and robust FR-based decoding for relevant classes of curves, including those with telescopic semigroups and semigroups at infinity.

Abstract

We present an algorithm to compute the Weierstrass semigroup at a point P together with functions for each value in the semigroup, provided P is the only branch at infinity of a singular plane model for the curve. As a byproduct, the method also provides us with a basis for the spaces L(mP) and the computation of the Feng-Rao distance for the corresponding array of geometric Goppa codes. A general computation of the Feng-Rao distance is also obtained. Everything can be applied to the decoding problem by using the majority scheme of Feng and Rao.

Computing Weierstrass semigroups and the Feng-Rao distance from singular plane models

TL;DR

The paper develops an algorithm to compute the Weierstrass semigroup at a point on a curve with a singular plane model having a single branch at infinity, using Abhyankar–Moh theory and approximate roots, complemented by a triangulation step from the integral closure. This yields not only a basis for spaces but also a practical route to compute the Feng–Rao distance for corresponding one-point AG codes, with a general Apéry-system framework that yields for arbitrary semigroups and refinements for symmetric and infinity-related semigroups. The results enable improved decoding of algebraic geometry codes through the Feng–Rao majority scheme and strengthen the bridge between singularity theory and coding theory by providing constructive semigroup descriptions and pole-order functions. Practically, the approach supports efficient code construction, explicit bases for , and robust FR-based decoding for relevant classes of curves, including those with telescopic semigroups and semigroups at infinity.

Abstract

We present an algorithm to compute the Weierstrass semigroup at a point P together with functions for each value in the semigroup, provided P is the only branch at infinity of a singular plane model for the curve. As a byproduct, the method also provides us with a basis for the spaces L(mP) and the computation of the Feng-Rao distance for the corresponding array of geometric Goppa codes. A general computation of the Feng-Rao distance is also obtained. Everything can be applied to the decoding problem by using the majority scheme of Feng and Rao.

Paper Structure

This paper contains 10 sections, 12 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Weierstrass semigroups and AG codes
  3. Computing Weierstrass semigroups
  4. Semigroups of plane curves with only one branch at infinity
  5. Approximate roots
  6. Application to codes
  7. Computing the Feng-Rao distance
  8. Apéry systems and Feng-Rao distance
  9. Symmetric semigroups
  10. Semigroups at infinity

Key Result

Proposition 3.1

Let $A={\cal O}_{\chi}({\cal C})\;$ and $\;B={\cal O}_{\Upsilon}({\cal Z})$ be the respective affine coordinate $\hbox{${\rm I} \! {\rm F}$}$-algebras of the curves ${\cal C}$ and ${\cal Z}$; then one has where $Q$ ranges over all the closed points of ${\cal Z}$ and $\delta_{Q}({\cal Z})= dim_{{ I} ! { F}}(\overline{\cal O}_{\Upsilon,Q}/{\cal O}_{\Upsilon,Q})$, $\overline{\cal O}_{\Upsilon,Q}$ be

Theorems & Definitions (26)

  • Proposition 3.1
  • Remark 3.3
  • Proposition 3.5: Criterion for having only one branch at infinity
  • Proposition 3.6: Abhyankar-Moh theorem
  • Remark 3.7
  • Remark 3.8
  • Remark 3.9
  • Example 3.10
  • Remark 3.11
  • Remark 3.12
  • ...and 16 more