Hermitian codes from higher degree places

Matthews and Michel investigated the minimum distances in certain algebraic-geometry codes arising from a higher degree place $P$. In terms of the Weierstrass gap sequence at $P$, they proved a bound that gives an improvement on the designed minimum distance. In this paper, we consider those of such codes which are constructed from the Hermitian function field. We determine the Weierstrass gap sequence $G(P)$ where $P$ is a degree 3 place, and compute the Matthews and Michel bound with the corresponding improvement. We show more improvements using a different approach based on geometry. We also compare our results with the true values of the minimum distances of Hermitian 1-point codes, as well as with estimates due Xing and Chen

An Introduction to Algebraic Geometry codes

We present an introduction to the theory of algebraic geometry codes. Starting from evaluation codes and codes from order and weight functions, special attention is given to one-point codes and, in particular, to the family of Castle codes

AG codes from the second generalization of the GK maximal curve

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