Horacio Tapia Recillas

arXiv2020

On the dimension of ideals in group algebras, and group codes

Several relations and bounds for the dimension of principal ideals in group algebras are determined by analyzing minimal polynomials of regular representations. These results are used in the two last sections. First, in the context of semisimple group algebras, to compute, for any abelian code, an element with Hamming weight equal to its dimension. Finally, to get bounds on the minimum distance of certain MDS group codes. A relation between a class of group codes and MDS codes is presented. Examples illustrating the main results are provided.

cs.IT↗

arXiv2018

$\mathbb{F}_{q}[G]$-modules and $G$-invariant codes

If $\mathbb{F}_{q}$ is a finite field, $C$ is a vector subspace of $\mathbb{F}_{q}^{n}$ (linear code), and $G$ is a subgroup of the group of linear automorphisms of $\mathbb{F}_{q}^{n}$, $C$ is said to be $G$-invariant if $g(C)=C$ for all $g\in G$. A solution to the problem of computing all the $G$-invariant linear codes $C$ of $\mathbb{F}_{q}^{n}$ is offered. This will be referred as the invariance problem. When $n=|G|t$, we determine conditions for the existence of an isomorphism of $\mathbb{F}_{q}[G]$-modules between $\mathbb{F}_{q}^{n}$ and $\mathbb{F}_{q}[G]\times \cdots \times \mathbb{F}_{q}[G]$ ($t$-times), that preserves the Hamming weight. This reduces the invariance problem to the determination of the $\mathbb{F}_{q}[G]$-submodules of $\mathbb{F}_{q}[G]\times \cdots \times \mathbb{F}_{q}[G]$ ($t$-times). The concept of Gaussian binomial coefficient for semisimple $\mathbb{F}_{q}[G]$-modules, which is useful for counting $G$-invariant codes, is introduced. Finally, a systematic way to compute all the $G$-invariant linear codes $C\subseteq \mathbb{F}_{q}^{n}$ is provided, when $(|G|,q)=1$.

cs.IT↗