# AG codes from the second generalization of the GK maximal curve

## TL;DR

This paper studies the algebraic geometry codes derived from the second generalization of GK maximal curves, providing new insights into their structure, properties, and potential for improved coding performance.

## Contribution

It determines the Weierstrass semigroup structure, proves these points are Weierstrass points, and constructs AG and quantum codes with improved parameters.

## Key findings

- Weierstrass semigroup structure determined for $ ext{GK}_{2,n}$
- Points on $ ext{GK}_{2,n}$ are Weierstrass points
- Constructed AG and quantum codes with better parameters

## Abstract

The second generalized GK maximal curves $\mathcal{GK}_{2,n}$ are maximal curves over finite fields with $q^{2n}$ elements, where $q$ is a prime power and $n \geq 3$ an odd integer, constructed by Beelen and Montanucci. In this paper we determine the structure of the Weierstrass semigroup $H(P)$ where $P$ is an arbitrary $\mathbb{F}_{q^2}$-rational point of $\mathcal{GK}_{2,n}$. We show that these points are Weierstrass points and the Frobenius dimension of $\mathcal{GK}_{2,n}$ is computed. A new proof of the fact that the first and the second generalized GK curves are not isomorphic for any $n \geq 5$ is obtained. AG codes and AG quantum codes from the curve $\mathcal{GK}_{2,n}$ are constructed; in some cases, they have better parameters with respect to those already known.