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John H. Castillo

Publications and source records attributed to John H. Castillo.

14 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, $\Gamma(C)$, 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

$S_h$-sets and linear codes over $\mathbb{F}_q$

Let $(G,+)$ be an Abelian group. Given $h\in \mathbb{Z}^+$, a non-empty subset $A$ of $G$ is called an $S_h$-set if all the sums of $h$ distinct elements of $A$ are different. We extend the concept of $S_h$-set to a more general context in the context of finite vectorial spaces over finite fields. More precisely, a $\emptyset \neq A\subseteq \mathbb{F}_q^r$ is called an $S_h$-linear set if all the linear combinations of $h$ elements of $A$ are different. We establish a correspondence between $q$-ary linear codes and $S_h$-linear sets. This connection allow us to find lower bounds for the maximum size of $S_h$-sets in $\mathbb{F}_q^r$.

math.NT↗
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↗
arXiv2017

The set of $k$-units modulo $n$

Let $R$ be a ring with identity, $\mathcal{U}(R)$ the group of units of $R$ and $k$ a positive integer. We say that $a\in \mathcal{U}(R)$ is $k$-unit if $a^k=1$. Particularly, if the ring $R$ is $\mathbb{Z}_n$, for a positive integer $n$, we will say that $a$ is a $k$-unit modulo $n$. We denote with $\mathcal{U}_k(n)$ the set of $k$-units modulo $n$. By $\text{du}_k(n)$ we represent the number of $k$-units modulo $n$ and with $\text{rdu}_k(n)=\frac{ϕ(n)}{\text{du}_k(n)}$ the ratio of $k$-units modulo $n$, where $ϕ$ is the Euler phi function. Recently, S. K. Chebolu proved that the solutions of the equation $\text{rdu}_2(n)=1$ are the divisors of $24$. The main result of this work, is that for a given $k$, we find the positive integers $n$ such that $\text{rdu}_k(n)=1$. Finally, we give some connections of this equation with Carmichael's numbers and two of its generalizations: Knödel numbers and generalized Carmichael numbers.

math.NT↗
arXiv2014

$q$-pseudoprimality: A natural generalization of strong pseudoprimality

In this work we present a natural generalization of strong pseudoprime to base $b$, which we have called $q$-pseudoprime to base $b$. It allows us to present another way to define a Midy's number to base $b$ (overpseudoprime to base $b$). Besides, we count the bases $b$ such that $N$ is a $q$-probable prime base $b$ and those ones such that $N$ is a Midy's number to base $b$. Furthemore, we prove that there is not a concept analogous to Carmichael numbers to $q$-probable prime to base $b$ as with the concept of strong pseudoprimes to base $b$.

math.NT↗
arXiv2013

Lie nilpotency indices of symmetric elements under oriented involutions in group algebras

Let $G$ be a group and let $F$ be a field of characteristic different from 2. Denote by $(FG)^+$ the set of symmetric elements and by $\mathcal{U}^+(FG)$ the set of symmetric units, under an oriented classical involution of the group algebra $FG$. We give some lower and upper bounds on the Lie nilpotency index of $(FG)^+$ and the nilpotency class of $\mathcal{U}^+(FG)$.

math.RA↗
arXiv2012

Overpseudoprimes, and Mersenne and Fermat numbers as primover numbers

We introduce a new class of pseudoprimes-so called "overpseudoprimes to base $b$", which is a subclass of strong pseudoprimes to base $b$. Denoting via $|b|_n$ the multiplicative order of $b$ modulo $n$, we show that a composite $n$ is overpseudoprime if and only if $|b|_d$ is invariant for all divisors $d>1$ of $n$. In particular, we prove that all composite Mersenne numbers $2^{p}-1$, where $p$ is prime, are overpseudoprime to base 2 and squares of Wieferich primes are overpseudoprimes to base 2. Finally, we show that some kinds of well known numbers are overpseudoprime to a base $b$.

math.NT↗
arXiv2012

Pseudoprimes stronger than strong pseudoprimes

We introduce a new class of pseudoprimes. In this work we characterize Midy pseudoprimes, give some of their properties and established interesting connections with other known pseudoprimes, in particular we show that every divisor of a Midy pseudoprime is either a prime or a Midy pseudoprime and in the last case it is a strong pseudoprime.

math.NT↗
arXiv2012

Structure of associated sets to Midy's Property

Let $b$ be a positive integer greater than 1, $N$ a positive integer relatively prime to $b$, $ |b|_{N}$ the order of $b$ in the multiplicative group $% \mathbb{U}_{N}$ of positive integers less than $N$ and relatively primes to $% N,$ and $x\in\mathbb{U}_{N}$. It is well known that when we write the fraction $\frac{x}{N}$ in base $b$, it is periodic. Let $d,\,k$ be positive integers with $% d\geq2$ and such that $|b|_{N}=kd$ and $\frac{x}{N}=0.% bar{a_{1}a_{2}...a_{|b|_{N}}}$ with the bar indicating the period and $a_{i}$ are digits in base $b$. We separate the period ${a_{1}a_{2}... a_{|b|_{N}}}$ in $d$ blocks of length $k$ and let $ A_{j}=[a_{(j-1)k+1}a_{(j-1)k+2}...a_{jk}]_{b} $ be the number represented in base $b$ by the $j-th$ block and $% S_{d}(x)=\sum\limits_{j=1}^{d}A_{j}$. If for all $x\in\mathbb{U}_{N}$, the sum $S_{d}(x)$ is a multiple of $b^{k}-1$ we say that $N$ has the Midy's property for $b$ and $d$. In this work we present some interesting properties of the set of positive integers $d$ such that $N$ has the Midy's property for $b$ and $d$.

math.NT↗