# Weierstrass Semigroup and Pure Gaps at several points on the $GK$ curve

## TL;DR

This paper characterizes the Weierstrass semigroup at multiple points on the GK curve, identifies pure gaps, and applies these findings to construct algebraic geometry codes with improved parameters.

## Contribution

It provides explicit descriptions of the Weierstrass semigroup and pure gaps at multiple points on the GK curve, enabling better code construction.

## Key findings

- Explicit description of Weierstrass semigroup at multiple points
- Conditions for identifying pure gaps on the GK curve
- Construction of algebraic geometry codes with improved parameters

## Abstract

We determine the Weierstrass semigroup $H(P_{\infty}, P_{1}, \ldots , P_{m})$ at several points on the $GK$ curve. In addition, we present conditions to find pure gaps on the set of gaps $G(P_{\infty}, P_{1}, \ldots , P_{m})$. Finally, we apply the results to obtain AG codes with good relative parameters.