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

Manuel González Sarabia; Carlos Rentería Márquez; Eliseo Sarmiento Rosales

Vol. 18 No. 2 (2013)

Regular Articles

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

Manuel González Sarabia
Carlos Rentería Márquez
Eliseo Sarmiento Rosales

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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.

Keywords

finite fields, regularity index, minimum distance, parameterized codes, embedded sets, complete graphs

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