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arXiv · 1901.08897

AG codes from the second generalization of the GK maximal curve

Maria Montanucci·Vicenzo Pallozzi Lavorante
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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.

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Maria Montanucci, Vicenzo Pallozzi Lavorante. 2019-01-25. AG codes from the second generalization of the GK maximal curve. https://arxiv.org/abs/1901.08897

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