Weierstrass Semigroups from Kummer Extensions

TL;DR

This paper explicitly determines the minimal generating sets of Weierstrass semigroups for certain Kummer extensions, aiding the construction of better algebraic geometry codes.

Contribution

It extends existing results by explicitly characterizing Weierstrass semigroups in general Kummer extensions using Matthews' techniques.

Findings

Explicit minimal generating sets obtained for various Kummer extensions

Extended previous specific cases to more general settings

Illustrative examples demonstrating the results

Abstract

The Weierstrass semigroups and pure gaps can be helpful in constructing codes with better parameters. In this paper, we investigate explicitly the minimal generating set of the Weierstrass semigroups associated with several totally ramified places over arbitrary Kummer extensions. Applying the techniques provided by Matthews in her previous work, we extend the results of specific Kummer extensions studied in the literature. Some examples are included to illustrate our results.