arXiv subjects

Alexander Holguín-Villa

Publications and source records attributed to Alexander Holguín-Villa.

6 recordsLinked to original sources

arXiv2026

Operations on binary linear codes and their associated graphs

This manuscript addresses a connection between operations in coding theory and graph theory, as well as some operations in both areas. The concept of the Hasse diagram of a binary linear code, $Γ(C)$, is recall; this concept visually represents the cosets of a binary linear code with a partial ordering. The main objetive is to give a characterization of the graphs corresponding to: extended codes, punctured codes, and the direct sum of two codes, using the graph of the given binary linear code as a starting point. Relationships are established between the graphs of code operations and particular operations of graphs.

cs.IT↗

arXiv2025

Cayley unitary elements in group algebras under oriented involutions

Let $\mathbf{F}$ be a real extension of $\mathbb{Q}$, $G$ a finite group and $\mathbf{F}G$ its group algebra. Given both a group homomorphism $σ:G\rightarrow \{\pm1\}$ (called an orientation) and a group involution $^\ast:G \rightarrow G$ such that $gg^\ast\in N=ker(σ)$, an oriented group involution $\circledast$ of $\mathbf{F}G$ is defined by $α=\sum_{g\in G}α_{g}g \mapsto α^\circledast=\sum_{g\in G}α_{g}σ(g)g^{\ast}$. In this paper, in case the involution on $G$ is the classical one, $x\mapsto x^{-1}$, $β=x+x^{-1}$ is a skew-symmetric element in $\mathbf{F}G$ such that $1+β$ is invertible, for $x\in G$ with $σ(x)=-1$, we consider Cayley unitary elements built out of $β$. We prove that the coefficients of $(1+β)^{-1}$ involve an interesting sequence which is a Fibonacci-like sequence.

math.RA↗

arXiv2023

Group identities on symmetric units under oriented involutions in group algebras

Let $\mathbb{F}G$ denote the group algebra of a locally finite group $G$ over the infinite field $\mathbb{F}$ with $char(\mathbb{F})\neq 2$, and let $\circledast:\mathbb{F}G\rightarrow \mathbb{F}G$ denote the involution defined by $α=Σα_{g}g \mapsto α^\circledast=Σα_{g}σ(g)g^{\ast}$, where $σ:G\rightarrow \{\pm1\}$ is a group homomorphism (called an orientation) and $\ast$ is an involution of the group $G$. In this paper we prove, under some assumptions, that if the $\circledast$-symmetric units of $\mathbb{F}G$ satisfies a group identity then $\mathbb{F}G$ satisfies a polynomial identity, i.e., we give an affirmative answer to a Conjecture of B. Hartley in this setting. Moreover, in the case when the prime radical $η(\mathbb{F}G)$ of $\mathbb{F}G$ is nilpotent we characterize the groups for which the symmetric units $\mathcal{U}^+(\mathbb{F}G)$ do satisfy a group identity.

math.RA↗

arXiv2022

On Duo, Reversible and Symmetric Group Rings

Let $RG$ denote the group ring of the torsion group $G$ over a commutative ring $R$ with identity. In this paper we present proofs of some statements that appear without to be proved in the literature. We establish the valid implications between the ring-theoretic conditions duo, reversible, SI property and symmetric in the setting of group rings. We further show that if the group ring $RG$ possesses any of these properties, then $G$ is a Hamiltonian group and the characteristic of $R$ is either $0$ or $2$. Moreover, we characterize the same properties in group rings $RG$ in the following cases: ($1)$ $RG$ is a semi-simple group ring and ($2$) $R$ is a semi-simple ring and $G$ any group.

math.RA↗

arXiv2022

A graphical representation of binary linear codes

A binary $[n,k]$-linear code $\mathcal{C}$ is a $k$-dimensional subspace of $\mathbb{F}_2^n$. For $\boldsymbol{x}\in \mathbb{F}_2^n$, the set $\boldsymbol{x}+\mathcal{C}$ is a coset of $\mathcal{C}$. In this work we study a partial ordering on the set of cosets of a binary linear code $\mathcal{C}$ of length $n$ and we construct a graph using the orphan structure of this code.

cs.IT↗

arXiv2019

Normal group algebras

Let $\mathbb{F}G$ denote the group algebra of the group $G$ over the field $\mathbb{F}$ with $char(\mathbb{F})\neq 2$. Given both a homomorphism $σ:G\rightarrow \{\pm1\}$ and a group involution $\ast: G\rightarrow G$, an oriented involution of $\mathbb{F}G$ is defined by $α=Σα_{g}g \mapsto α^\circledast=Σα_{g}σ(g)g^{\ast}$. In this paper, we determine the conditions under which the group algebra $\mathbb{F}G$ is normal, that is, conditions under which $\mathbb{F}G$ satisfies the $\circledast$-identity $αα^\circledast=α^\circledastα$. We prove that $\mathbb{F}G$ is normal if and only if the set of symmetric elements under $\circledast$ is commutative.

math.RA↗