AG codes from the second generalization of the GK maximal curve

The second generalized GK maximal curves ${\mathcal{G}\mathcal{K}}_{2,n}$ are maximal curves over finite fields with $q^{2n}$ elements, where $q$ is a prime power and $n\ge 3$ 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 ${\mathcal{G}\mathcal{K}}_{2,n}$. We show that these points are Weierstrass points and the Frobenius dimension of ${\mathcal{G}\mathcal{K}}_{2,n}$ is computed. A new proof of the fact that the first and the second generalized GK curves are not isomorphic for any $n\ge 5$ is obtained. AG codes and AG quantum codes from the curve ${\mathcal{G}\mathcal{K}}_{2,n}$ are constructed; in some cases, they have better parameters with respect to those already known.