One- and Two-Point Codes over Kummer Extensions

TL;DR

This paper calculates Weierstrass semigroups at specific places in Kummer extensions and uses these results to construct algebraic geometric codes with improved parameters.

Contribution

It provides explicit computations of Weierstrass semigroups at one and two totally ramified places in Kummer extensions, enabling better code construction.

Findings

Explicit formulas for Weierstrass semigroups at ramified places

Construction of algebraic geometric codes with improved parameters

Enhanced understanding of Kummer extension structures

Abstract

We compute the Weierstrass semigroup at one totally ramified place for Kummer extensions defined by $y^m=f(x)^{\lambda}$ where $f(x)$ is a separable polynomial over $\mathbb{F}_q$. In addition, we compute the Weierstrass semigroup at two certain totally ramified places. We then apply our results to construct one- and two-point algebraic geometric codes with good parameters.