arXiv subjects

Juan Jacobo Simón

Publications and source records attributed to Juan Jacobo Simón.

14 recordsLinked to original sources

arXiv2026

Graded Equivalence for Graded Idempotent Rings

In this paper, we extend the study of graded equivalences to the case of general idempotent graded rings. We prove that the existence of a graded equivalence between two categories of graded torsion-free unital modules may be characterized by the existence of a Morita context with surjective trace maps. As an application of our results we relate certain lattices of graded submodules and graded ideals of graded equivalent garded rings and give some properties invariant under graded equivalences.

math.RA↗

arXiv2026

Homology and cohomology of crossed products by inverse monoid actions and Steinberg algebras

Given a unital action $θ$ of an inverse monoid $S$ on an algebra $A$ over a filed $K$ we produce (co)homology spectral sequences which converge to the Hochschild (co)homology of the crossed product $A\rtimes_θS$ with values in a bimodule over $A\rtimes_θS$. The spectral sequences involve a new kind of (co)homology of the inverse monoid $S,$ which is based on $KS$-modules. The spectral sequences take especially nice form, when $(A\rtimes_θS)^e $ is flat as a left (homology case) or right (cohomology case) $A^e$-module, involving also the Hochschild (co)homology of $A.$ Same nice spectral sequences are also obtained if $K$ is a commutative ring, over which $A$ is projective, and $S$ is $E$-unitary. We apply our results to the Steinberg algebra $A_K(\mathscr{G})$ over a field $K$ of an ample groupoid $\mathscr{G},$ whose unit space $\mathscr{G} ^{(0)}$ is compact. In the homology case our spectral sequence collapses on the $p$-axis, resulting in an isomorphism between the Hochschild homology of $A_K(\mathscr{G})$ with values in an $A_K(\mathscr{G})$-bimodule $M$ and the homology of the inverse semigroup of the compact open bisections of $\mathscr{G}$ with values in the invariant submodule of $M.$

math.RA↗

arXiv2025

Permutation decoding of first-order Generalized Reed-Muller codes

In [4] we describe a variation of the classical permutation decoding algorithm that can be applied to any binary affine-invariant code; in particular, it can be applied to first-order Reed-Muller codes successfully. In this paper we study how to implement it for the family of first-order Generalized Reed-Muller codes. Then, we give examples which show that we improve the number of errors we can correct in comparison with the known results for this family of codes. Finally, we deal, from a probabilistic point of view, with the problem of determining when the algorithm only needs to use a smaller PD-like set.

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arXiv2024

Apparent Distance and a Notion of BCH Multivariate Codes

This paper is devoted to studying two main problems: 1) computing the apparent distance of an Abelian code and 2) giving a notion of Bose, Ray-Chaudhuri, Hocquenghem (BCH) multivariate code. To do this, we first strengthen the notion of an apparent distance by introducing the notion of a strong apparent distance; then, we present an algorithm to compute the strong apparent distance of an Abelian code, based on some manipulations of hypermatrices associated with its generating idempotent. Our method uses less computations than those given by Camion and Sabin; furthermore, in the bivariate case, the order of computation complexity is reduced from exponential to linear. Then, we use our techniques to develop a notion of a BCH code in the multivariate case, and we extend most of the classical results on cyclic BCH codes. Finally, we apply our method to the design of Abelian codes with maximum dimension with respect to a fixed apparent distance and a fixed length.

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arXiv2024

Cyclic and BCH Codes whose Minimum Distance Equals their Maximum BCH bound

In this paper we study the family of cyclic codes such that its minimum distance reaches the maximum of its BCH bounds. We also show a way to construct cyclic codes with that property by means of computations of some divisors of a polynomial of the form X^n-1. We apply our results to the study of those BCH codes C, with designed distance delta, that have minimum distance d(C)= delta. Finally, we present some examples of new binary BCH codes satisfying that condition. To do this, we make use of two related tools: the discrete Fourier transform and the notion of apparent distance of a code, originally defined for multivariate abelian codes.

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arXiv2024

Constructions of Abelian Codes multiplying dimension of cyclic codes

In this note, we apply some techniques developed in [1]-[3] to give a particular construction of bivariate Abelian Codes from cyclic codes, multiplying their dimension and preserving their apparent distance. We show that, in the case of cyclic codes whose maximum BCH bound equals its minimum distance the obtained abelian code verifies the same property; that is, the strong apparent distance and the minimum distance coincide. We finally use this construction to multiply Reed-Solomon codes to abelian codes

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arXiv2024

An intrinsical description of group codes

A (left) group code of length n is a linear code which is the image of a (left) ideal of a group algebra via an isomorphism from FG to Fn which maps G to the standard basis of Fn. Many classical linear codes have been shown to be group codes. In this paper we obtain a criterion to decide when a linear code is a group code in terms of its intrinsical properties in the ambient space Fn, which does not assume an a priori group algebra structure on Fn. As an application we provide a family of groups (including metacyclic groups) for which every two-sided group code is an abelian group code. It is well known that Reed-Solomon codes are cyclic and its parity check extensions are elementary abelian group codes. These two classes of codes are included in the class of Cauchy codes. Using our criterion we classify the Cauchy codes of some lengths which are left group codes and the possible group code structures on these codes.

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arXiv2024

A new approach to the Berlekamp-Massey-Sakata Algorithm. Improving Locator Decoding

We study the problem of the computation of Groebner basis for the ideal of linear recurring relations of a doubly periodic array. We find a set of indexes such that, along with some conditions, guarantees that the set of polynomials obtained at the last iteration in the Berlekamp-Massey-Sakata algorithm is exactly a Groebner basis for the mentioned ideal. Then, we apply these results to improve locator decoding in abelian codes.

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arXiv2024

Information sets from defining sets for Reed-Muller codes of first and second order

Reed-Muller codes belong to the family of affine-invariant codes. As such codes they have a defining set that determines them uniquely, and they are extensions of cyclic group codes. In this paper we identify those cyclic codes with multidimensional abelian codes and we use the techniques introduced in \cite{BS} to construct information sets for them from their defining set. For first and second order Reed-Muller codes, we describe a direct method to construct information sets in terms of their basic parameters.

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arXiv2023

New advances in permutation decoding of first-order Reed-Muller codes

In this paper we describe a variation of the classical permutation decoding algorithm that can be applied to any affine-invariant code with respect to certain type of information sets. In particular, we can apply it to the family of first-order Reed-Muller codes with respect to the information sets introduced in [2]. Using this algortihm we improve considerably the number of errors we can correct in comparison with the known results in this topic.

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arXiv2020

Decoding up to 4 errors in Hyperbolic-like Abelian Codes by the Sakata Algorithm

We deal with two problems related with the use of the Sakata's algorithm in a specific class of bivariate codes. The first one is to improve the general framework of locator decoding in order to apply it on such abelian codes. The second one is to find a set of indexes oF the syndrome table such that no other syndrome contributes to implement the BMSa and, moreover, any of them may be ignored \textit{a priori}. In addition, the implementation on those indexes is sufficient to get the Groebner basis; that is, it is also a termination criterion.

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arXiv2020

Globalization of partial cohomology of groups

We study the relations between partial and global group cohomology with values in a commutative unital ring $\mathcal{A}$. In particular, for a unital partial action of a group $G$ on $\mathcal{A}$, such that $\mathcal{A}$ is a direct product of commutative indecomposable rings, we show that any partial $n$-cocycle of $G$ with values in $\mathcal{A}$ is globalizable.

math.RA↗

arXiv2019

Globalization of group cohomology in the sense of Alvares-Alves-Redondo

Recently E. R. Alvares, M. M. Alves and M. J. Redondo introduced a cohomology for a group $G$ with values in a module over the partial group algebra $K_{\mathrm{par}}(G)$, which is different from the partial group cohomology defined earlier by the first two named authors of the present paper. Given a unital partial action $α$ of $G$ on a (unital) algebra $\mathcal{A}$ we consider $\mathcal{A}$ as a $K_{\mathrm{par}}(G)$-module in a natural way and study the globalization problem for the cohomology in the sense of Alvares-Alves-Redondo with values in $\mathcal{A}$. The problem is reduced to an extendibility property of cocycles. Furthermore, assuming that $\mathcal{A}$ is a product of blocks, we prove that any cocycle is globalizable, and globalizations of cohomologous cocycles are also cohomologous. As a consequence we obtain that the Alvares-Alves-Redondo cohomology group $H_{par}^n(G,\mathcal{A})$ is isomorphic to the usual cohomology group $H^n(G,\mathcal{M}(\mathcal{B}))$, where $\mathcal{M}(\mathcal{B})$ is the multiplier algebra of $\mathcal{B}$ and $\mathcal{B}$ is the algebra under the enveloping action of $α$.

math.RA↗

arXiv2011

Information sets from defining sets in abelian codes

We describe a technique to construct a set of check positions (and hence an information set) for every abelian code solely in terms of its defining set. This generalizes that given by Imai in \cite{Imai} in the case of binary TDC codes.

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