On AG codes from a generalization of the Deligne-Lustzig curve of Suzuki type

In this paper, Algebraic-Geometric (AG) codes and quantum codes associated to a family of curves which comprises the famous Suzuki curve are investigated. The Weierstrass semigroup at some rational point is computed. Notably, each curve in the family turn out to be a Castle curve over some finite field, and a weak Castle curve over its extensions. This is a relevant feature when codes constructed from the curve are considered

Curves with more than one inner Galois point

Let $\mathcal{C}$ be an irreducible plane curve of $\text{PG}(2,\mathbb{K})$ where $\mathbb{K}$ is an algebraically closed field of characteristic $p\geq 0$. A point $Q\in \mathcal{C}$ is an inner Galois point for $\mathcal{C}$ if the projection $\pi_Q$ from $Q$ is Galois. Assume that $\mathcal{C}$ has two different inner Galois points $Q_1$ and $Q_2$, both simple. Let $G_1$ and $G_2$ be the respective Galois groups. Under the assumption that $G_i$ fixes $Q_i$, for $i=1,2$, we provide a complete classification of $G=\langle G_1,G_2 \rangle$ and we exhibit a curve for each such $G$. Our proof relies on deeper results from group theory

Complete (q+1)(q+1)-arcs in PG(2,Fq6)\mathrm{PG}(2,\mathbb{F}_{q^6}) from the Hermitian curve

We prove that, if $q$ is large enough, the set of the $\mathbb{F}_{q^6}$-rational points of the Hermitian curve is a complete $(q+1)$-arc in $\mathrm{PG}(2,\mathbb{F}_{q^6})$, addressing an open case from a recent paper by Korchm\'aros, Sz\H{o}nyi and Nagy. An algebraic approach based on the investigation of some algebraic varieties attached to the arc is used

Minimal codewords in Norm-Trace codes

In this paper, we consider the affine variety codes obtained evaluating the polynomials $by=a_kx^k+\dots+a_1x+a_0$, $b,a_i\in\mathbb{F}_{q^r}$, at the affine \F_{q^r}-rational points of the Norm-Trace curve. In particular, we investigate the weight distribution and the set of minimal codewords. Our approach, which uses tools of algebraic geometry, is based on the study of the absolutely irreducibility of certain algebraic varieties

New sextics of genus 6 and 10 attaining the Serre bound

We provide new examples of curves of genus 6 or 10 attaining the Serre bound. They all belong to the family of sextics introduced in [19] as a a generalization of the Wiman sextics [36] and Edge sextics [9]. Our approach is based on a theorem by Kani and Rosen which allows, under certain assumptions, to fully decompose the Jacobian of the curve. With our investigation we are able to update several entries in \url{http://www.manypoints.org} ([35])

Two-point AG codes from one of the Skabelund maximal curves

In this paper, we investigate two-point Algebraic Geometry codes associated to the Skabelund maximal curve constructed as a cyclic cover of the Suzuki curve. In order to estimate the minimum distance of such codes, we make use of the generalized order bound introduced by P. Beelen and determine certain two-point Weierstrass semigroups of the curve.Comment: 15 page