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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来自 GK 极大曲线的第二次推广的 AG 代码

第二个广义 GK 极大曲线 $\mathcal{GK}_{2,n}$ 是具有 $q^{2n}$ 元素的有限域上的极大曲线，其中 $q$ 是质数幂，$n \geq 3$一个奇数，由 Beelen 和 Montanucci 构造。在本文中，我们确定了 Weierstrass 半群 $H(P)$ 的结构，其中 $P$ 是 $\mathcal{GK}_{2 的任意 $\mathbb{F}_{q^2}$-有理点,n}$。我们证明这些点是 Weierstrass 点，并且计算了 $\mathcal{GK}_{2,n}$ 的 Frobenius 维数。获得了第一和第二广义 GK 曲线对于任何 $n \geq 5$ 都不是同构的事实的新证明。构造曲线$\mathcal{GK}_{2,n}$中的AG码和AG量子码；在某些情况下，与已知参数相比，它们具有更好的参数。