Parameterized codes over some embedded sets and their applications to complete graphs

Abstract

Let $K$ be a finite field, let $X \subset \mathbb{P}^{m-1}$ and $X' \subset \mathbb{P}^{r-1}$, with $r<m$, be two algebraic toric sets parameterized by some monomials in such a way that $X'$ is embedded in $X$. We describe the relations among the main parameters of the corresponding parameterized linear codes of order $d$ associated to $X$ and $X'$ by using some tools from commutative algebra and algebraic geometry.
We also find the regularity index in the case of toric sets parameterized by the edges of a complete graph. Finally, we give some bounds for the minimum distance of the linear codes associated to complete graphs.