José Joaquín Bernal | Universidad de Murcia (original) (raw)

Papers by José Joaquín Bernal

Cornell University - arXiv, Oct 1, 2016

In this paper we develop a technique to extend any bound for cyclic codes constructed from its de... more In this paper we develop a technique to extend any bound for cyclic codes constructed from its defining sets (ds-bounds) to abelian (or multivariate) codes. We use this technique to improve the searching of new bounds for abelian codes.

IEEE Transactions on Information Theory, 2018

Reed-Muller codes belong to the family of affineinvariant codes. As such codes they have a defini... more Reed-Muller codes belong to the family of affineinvariant 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 [4] 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.

ArXiv, 2017

In this paper we develop a technique to extend any bound for the minimum distance of cyclic codes... more In this paper we develop a technique to extend any bound for the minimum distance of cyclic codes constructed from its defining sets (ds-bounds) to abelian (or multivariate) codes through the notion of mathbbB\mathbb{B}mathbbB-apparent distance. We use this technique to improve the searching for new bounds for the minimum distance of abelian codes. We also study conditions for an abelian code to verify that its mathbbB\mathbb{B}mathbbB-apparent distance reaches its (true) minimum distance. Then we construct some tables of such codes as an application

Coding Theory and Applications, 2017

As affine-invariant codes, Reed-Muller codes are extension of cyclic group codes and they have a ... more As affine-invariant codes, Reed-Muller codes are extension of cyclic group codes and they have a defining set that determines them uniquely. In this paper we identify those cyclic codes with multidimensional abelian codes and we use the techniques introduced in [3] to construct information sets for first and second order Reed-Muller codes from its defining set.

We deal with two problems related with the use of the Sakata's algorithm in a specific class ... more 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.

Mathematics in Computer Science, 2019

In this note, we apply some techniques developed in

IEEE Transactions on Information Theory, 2019

In this paper we develop a technique to extend any bound for the minimum distance of cyclic codes... more In this paper we develop a technique to extend any bound for the minimum distance of cyclic codes constructed from its defining sets (ds-bounds) to abelian (or multivariate) codes through the notion of B-apparent distance. We also study conditions for an abelian code to verify that its B-apparent distance reaches its (true) minimum distance. Then we construct some codes as an application.

Advances in Mathematics of Communications, 2016

In this paper we study the family of cyclic codes such that its minimum distance reaches the maxi... more 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 δ, that have minimum distance d(C) = δ. 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.

IEEE Transactions on Information Theory, 2016

This paper is devoted to studying two main problems: 1) computing the apparent distance of an Abe... more 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.

ACM Communications in Computer Algebra, 2015

In this extended abstract, we use the techniques of computation of the minimum apparent distance ... more In this extended abstract, we use the techniques of computation of the minimum apparent distance of a hypermatrix given in [2] in order to develop a notion of BCH bound and BCH code in the multivariate case. Then we extend the most classical results in BCH codes to the multivariate case and we show how to construct abelian codes with maximum dimension with respect to prefixed bounds for their minimum distance. Keywords Minimum apparent distance of a hypermatrix, Apparent distance of an abelian code, Multivariable BCH bound, Multivarible BCH code.

2013 IEEE Information Theory Workshop (ITW), 2013

ABSTRACT

2010 IEEE Information Theory Workshop, 2010

Page 1. 2010 IEEE Information Theory Workshop - ITW 2010 Dublin Information sets for abelian code... more Page 1. 2010 IEEE Information Theory Workshop - ITW 2010 Dublin Information sets for abelian codes José Joaquın Bernal and Juan Jacobo Simón Departamento de Matemáticas Universidad de Murcia 30100 Murcia, Spain Email: {josejoaquin.bernal, jsimon}@um.es ...

Lecture Notes in Computer Science, 2008

Page 1. How to Know if a Linear Code Is a Group Code? * José Joaquın Bernal, Ángel del Rıo, and J... more Page 1. How to Know if a Linear Code Is a Group Code? * José Joaquın Bernal, Ángel del Rıo, and Juan Jacobo Simón Departamento de Matemáticas, Universidad de Murcia, Espana josejoaquin.bernal@alu.um.es, adelrio@um.es, jsimon@um.es Abstract. ...

Lecture Notes in Computer Science, 2009

We show that an affine-invariant code C of length p m is not permutation equivalent to a cyclic c... more We show that an affine-invariant code C of length p m is not permutation equivalent to a cyclic code except in the obvious cases: m = 1 or C is either {0}, the repetition code or its dual.

IEEE Transactions on Information Theory, 2013

In [3], we introduced a technique to construct an information set for every semisimple abelian co... more In [3], we introduced a technique to construct an information set for every semisimple abelian code over an arbitrary field, solely in terms of its defining set. In this paper we apply the geometrical properties of those information sets to obtain sufficient conditions for a terror correcting abelian code to have a b-PD-set for every b ≤ t. These conditions are simply given in terms of the structure of the defining set of the code.

IEEE Transactions on Information Theory, 2011

We describe a technique to construct a set of check positions (and hence an information set) for ... more 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 [7] in the case of binary TDC codes.

Designs, Codes and Cryptography, 2009

A (left) group code of length n is a linear code which is the image of a (left) ideal of a group ... more 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 FG → F n which maps G to the standard basis of F n. 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 F n , which does not assume an "a priori" group algebra structure on F n. 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.

Designs, Codes and Cryptography, 2012

In [BS2] we introduced a technique to construct information sets for every semisimple abelian cod... more In [BS2] we introduced a technique to construct information sets for every semisimple abelian code by means of its defining set. This construction is a non trivial generalization of that given by H. Imai [Im] in the case of binary two-dimensional cyclic (TDC) codes. On the other hand, S. Sakata [Sak] showed a method for constructing information sets for binary TDC codes based on the computation of Groebner basis which agrees with the information set obtained by Imai. Later, H.Chabanne [Chab] presents a generalization of the permutation decoding algorithm for binary abelian codes by using Groebner basis, and as a part of his method he constructs an information set following the same ideas introduced by Sakata. In this paper we show that, in the general case of q-ary multidimensional abelian codes, both methods, that based on Groebner basis and that defined in terms of the defining sets, also yield the same information set.