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Two-point AG codes from one of the Skabelund maximal curves

Leonardo Landi, Marco Timpanella, Lara Vicino

TL;DR

This work studies duals of two-point Algebraic Geometry codes on the Skabelund maximal curve $\tilde{\mathcal{S}}_q$, a cyclic cover of the Suzuki curve, with a focus on bounding the minimum distance via Beelen's generalized order bound. It explicitly determines the two-point Weierstrass semigroup $H(P, P_\infty)$ for $\tilde{\mathcal{S}}_q$ by analyzing the ring of regular functions and the known semigroup at infinity, and then applies the order bound to obtain distance estimates for two-point codes of the form $C_L(D, aP + bP_\infty)^{\perp}$. Numerical results for $s=1$ ($q=8$, $q_0=2$) show that the generalized order bound frequently improves the Goppa bound and that two-point codes can achieve at least the same minimum distance as the best one-point codes of the same dimension. Overall, the paper demonstrates the effectiveness of the Beelen bound for maximal-curves-based AG codes and supplies practical minimum-distance data for Skabelund-derived codes, aiding code-design and optimization.

Abstract

In this paper, we investigate two-point Algebraic Geometry codes associated to the Skabelund maximal curve constructed as a cyclic cover of the Suzuki curve. In order to estimate the minimum distance of such codes, we make use of the generalized order bound introduced by P. Beelen and determine certain two-point Weierstrass semigroups of the curve.

Two-point AG codes from one of the Skabelund maximal curves

TL;DR

This work studies duals of two-point Algebraic Geometry codes on the Skabelund maximal curve , a cyclic cover of the Suzuki curve, with a focus on bounding the minimum distance via Beelen's generalized order bound. It explicitly determines the two-point Weierstrass semigroup for by analyzing the ring of regular functions and the known semigroup at infinity, and then applies the order bound to obtain distance estimates for two-point codes of the form . Numerical results for (, ) show that the generalized order bound frequently improves the Goppa bound and that two-point codes can achieve at least the same minimum distance as the best one-point codes of the same dimension. Overall, the paper demonstrates the effectiveness of the Beelen bound for maximal-curves-based AG codes and supplies practical minimum-distance data for Skabelund-derived codes, aiding code-design and optimization.

Abstract

In this paper, we investigate two-point Algebraic Geometry codes associated to the Skabelund maximal curve constructed as a cyclic cover of the Suzuki curve. In order to estimate the minimum distance of such codes, we make use of the generalized order bound introduced by P. Beelen and determine certain two-point Weierstrass semigroups of the curve.
Paper Structure (7 sections, 11 theorems, 36 equations, 1 figure)

This paper contains 7 sections, 11 theorems, 36 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Algebraic Geometry codes
  4. The generalized order bound
  5. The Skabelund curve Sq
  6. The two-point Weierstrass semigroup H(P, P_inf)
  7. Results and comparisons

Key Result

Proposition 2.4

Let $C_L(D, G)$ be an AG code with $D$ and $G$ as in Definition def:orderbound. Then the minimum distance $d^\prime$ of the dual code $C_L(D, G)^\perp$ satisfies the inequality $d^\prime \geq d(G)$.

Figures (1)

  • Figure 1: The two-point Weierstrass semigroup $H(P, P_\infty)$ of $\tilde{\mathcal{S}}_q$ for $s=1$, of period $\pi=65$. Only the pairs $(i, j) \in H(P, P_\infty)$ with $-2 \pi < i, j < 2 \pi$ are represented.

Theorems & Definitions (21)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3: OrderBound*Definition 6
  • Proposition 2.4
  • Remark 2.5
  • Definition 2.6
  • Proposition 2.7: BT06*Proposition 14, Proposition 17
  • Theorem 2.8: TwoPointCodes*Theorem 9
  • Corollary 2.9: TwoPointCodes*Corollary 10
  • Proposition 2.10
  • ...and 11 more