Alexis E. Almendras Valdebenito

arXiv2019

Lower and upper bounds for some generalized arcs

Let $\mathbb{F}_q$ be a field with $q$ elements. In this note, we study some generalized arcs, that is, sets of $\mathbb{F}_q$-points in the projective plane $\mathbb{P}^2(\mathbb{F}_q)$ such that no six of them are on a conic. First, we consider the geometric configurations of such generalized arcs for small values of $q$ and then we give some upper and lower bounds for the cardinality of complete generalized arcs, i.e. generalized arcs which are not contained in a bigger one.

math.AG↗

arXiv2018

A note on the duals of skew constacyclic codes

Let $\mathbb{F}_q$ be a finite field with $q$ elements and denote by $θ: \mathbb{F}_q\to\mathbb{F}_q$ an automorphism of $\mathbb{F}_q$. In this paper, we deal with skew constacyclic codes, that is, linear codes of $\mathbb{F}_q^n$ which are invariant under the action of a semi-linear map $Φ_{α,θ}:\mathbb{F}_q^n\to\mathbb{F}_q^n$, defined by $Φ_{α,θ}(a_0,...,a_{n-2}, a_{n-1}):=(αθ(a_{n-1}),θ(a_0),...,θ(a_{n-2}))$ for some $α\in\mathbb{F}_q\setminus\{0\}$ and $n\geq 2$. In particular, we study some algebraic and geometric properties of their dual codes and we give some consequences and research results on $1$-generator skew quasi-twisted codes and on MDS skew constacyclic codes.

cs.IT↗