Subcovers and codes on the $X_{n,r}$ curves

TL;DR

This paper constructs subcovers of certain algebraic curves, determines their Weierstrass semigroups, and develops new algebraic geometry codes that outperform previous records, including 108 improvements on MinT tables.

Contribution

It introduces new subcovers of the curves, computes their Weierstrass semigroups, and constructs improved algebraic geometry codes with record parameters.

Findings

New algebraic geometry codes with record parameters

108 improvements on MinT tables

Explicit determination of Weierstrass semigroups

Abstract

In this work, subcovers $\mathcal{X}_{n,r}^s$ of the curve $\mathcal{X}_{n,r}$ are constructed, the Weierstrass semigroup $H(P_\infty)$ at the point $P_\infty \in \mathcal{X}_{n,r}^s$ is determined and the corresponding one-point AG codes are investigated. Codes establishing new records on the parameters with respect to the previously known ones are discovered, and $108$ improvements on MinT tables are obtained.