F. J. Lobillo

University of Granada | UGR · Department of Algebra

A class of linear codes that extends classical Goppa codes to a non-commutative context is defined. An efficient decoding algorithm, based on the solution of a non-commutative key equation, is designed. We show how the parameters of these codes, when the alphabet is a finite field, may be adjusted to propose a McEliece-type cryptosystem.

A class of linear codes that extends classic Goppa codes to a non-commutative context is defined. An efficient decoding algorithm, based on the solution of a non-commutative key equation, is designed. We show how the parameters of these codes, when the alphabet is a finite field, may be adjusted to propose a McEliece-type cryptosystem.

The tensor product of one code endowed with the Hamming metric and one endowed with the rank metric is analyzed. This gives a code which naturally inherits the sum-rank metric. Specializing to the product of a cyclic code and a skew-cyclic code, the resulting code turns out to belong to the recently introduced family of cyclic-skew-cyclic codes. A...

We consider the algorithmic problem of computing a primitive idempotent of a central simple algebra over the field of rational functions over a finite field. The algebra is given by a set of structure constants. The problem is reduced to the computation of a division algebra Brauer equivalent to the central simple algebra. This division algebra is...

We design a decoding algorithm for linear codes over finite chain rings given by their parity check matrices. It is assumed that decoding algorithms over the residue field are known at each degree of the adic decomposition.

Differential Convolutional Codes with designed Hamming distance are defined, and an algebraic decoding algorithm, inspired by Peterson–Gorenstein–Zierler’s algorithm, is designed for them.

The tensor product of one code endowed with the Hamming metric and one endowed with the rank metric is analyzed. This gives a code which naturally inherits the sum-rank metric. Specializing to the product of a cyclic code and a skew-cyclic code, the resulting code turns out to belong to the recently introduced family of cyclic-skew-cyclic. A group...

We give necessary and sufficient conditions on an Ore extension A[x;σ,δ]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A[x;\sigma ,\delta ]$$\end{document}, where A is...

Finding the minimum distance of linear codes is an NP-hard problem. Traditionally, this computation has been addressed by means of the design of algorithms that find, by a clever exhaustive search, a linear combination of some generating matrix rows that provides a codeword with minimum weight. Therefore, as the dimension of the code or the size of...

In this paper, a Roos like bound on the minimum distance for skew cyclic codes over a general field is provided. The result holds in the Hamming metric and in the rank metric. The proofs involve arithmetic properties of skew polynomials and an analysis of the rank of parity-check matrices. For the rank metric case, a way to arithmetically construct...

We consider the algorithmic problem of computing a primitive idempotent of a central simple algebra over the field of rational functions over a finite field. The algebra is given by a set of structure constants. The problem is reduced to the computation of a division algebra Brauer equivalent to the central simple algebra. This division algebra is...

Some relevant notions in fuzzy set theory are those of triangular norm and conorm, and negation, which provide a systematic way of defining set-theoretic operations or, from other point of view, logical connectives. For instance, the majority of fuzzy implications are directly derived from these operators, so they play a prominent role in fuzzy con...

Finding the minimum distance of linear codes is an NP-hard problem. Traditionally, this computation has been addressed by means of the design of algorithms that find, by a clever exhaustive search, a linear combination of some generating matrix rows that provides a codeword with minimum weight. Therefore, as the dimension of the code or the size of...

In this paper, a Roos like bound on the minimum distance for skew cyclic codes over a general field is provided. The result holds in the Hamming metric and in the rank metric. The proofs involve arithmetic properties of skew polynomials and an analysis of the rank of parity-check matrices. For the rank metric case, a way to arithmetically construct...

With a small suitable modification, dropping the projectivity condition, we extend the notion of a Frobenius algebra to grant that a Frobenius algebra over a Frobenius commutative ring is itself a Frobenius ring. The modification introduced here also allows Frobenius finite rings to be precisely those rings which are Frobenius finite algebras over...

Differential Convolutional Codes with designed Hamming distance are defined, and an algebraic decoding algorithm, inspired by Peterson-Gorenstein-Zierler's algorithm, is designed for them.

We show that, for convolutional codes endowed with a cyclic structure, it is possible to define and compute two sequences of positive integers, called cyclic column and row distances, which present a more regular behavior than the classical column and row distance sequences. We then design an algorithm for the computation of the free distance based...

We give necessary and sufficient conditions on an Ore extension $A[x;\sigma,\delta]$, where $A$ is a finite dimensional algebra over a field $\mathbb{F}$, for being a Frobenius extension over the ring of commutative polynomials $\mathbb{F}[x]$. As a consequence, as the title of this paper highlights, we provide a negative answer to a problem stated...

With a small suitable modification, dropping the projectivity condition, we extend the notion of a Frobenius algebra to grant that a Frobenius algebra over a Frobenius commutative ring is itself a Frobenius ring. The modification introduced here also allows Frobenius finite rings to be precisely those rings which are Frobenius finite algebras over...

We design a heuristic method, a genetic algorithm, for the computation of an upper bound of the minimum distance of a linear code over a finite field. By the use of the row reduced echelon form, we obtain a permutation encoding of the problem, so that its space of solutions does not depend on the size of the base field or the dimension of the code....

We design a heuristic method, a genetic algorithm, for the computation of an upper bound of the minimum distance of a linear code over a finite field. By the use of the row reduced echelon form, we obtain a permutation encoding of the problem, so that its space of solutions does not depend on the size of the base field or the dimension of the code....

We show that, for cyclic convolutional codes, it is possible to compute a sequence of positive integers, called cyclic column distances, which presents a more regular behavior than the classical column distances sequence. We then design an algorithm for the computation of the free distance based on the calculation of this cyclic column distances se...

In this paper we deal with the problem of extending Zadeh's operators on fuzzy sets (FSs) to interval-valued (IVFSs), set-valued (SVFSs) and type-2 (T2FSs) fuzzy sets. Namely, it is known that seeing FSs as SVFSs, or T2FSs, whose membership degrees are singletons is not order-preserving. We then describe a family of lattice embeddings from FSs to S...

In this paper a framework to study the dual of skew cyclic codes is proposed. The transposed Hamming ring extensions are based in the existence of an anti-isomorphism of algebras between skew polynomial rings. Our construction is applied to left ideal convolutional codes, skew constacyclic codes and skew Reed-Solomon code, showing that the dual of...

In this paper we deal with the problem of extending Zadeh's operators on fuzzy sets (FSs) to interval-valued (IVFSs), set-valued (SVFSs) and type-2 (T2FSs) fuzzy sets. Namely, it is known that seeing FSs as SVFSs, or T2FSs, whose membership degrees are singletons is not order-preserving. We then describe a family of lattice embeddings from FSs to S...

The use of skew polynomial rings allows to endow linear codes with cyclic structures which are not cyclic in the classical (commutative) sense. Whenever these skew cyclic structures are carefully chosen, some control over the Hamming distance is gained, and it is possible to design efficient decoding algorithms. In this paper, we give a version of...

This paper deals with cash management for bank branches, under the assumption that branches have a role to play in the improvement of global bank institution performance. In the current scenario of unprecedented pressure amongst banks to keep costs under control, our contribution is the design of a sound and low-cost algorithm to optimize branch ca...

We design a non-commutative version of the Peterson-Gorenstein-Zierler decoding algorithm for a class of codes that we call skew RS codes. These codes are left ideals of a quotient of a skew polynomial ring, which endow them of a sort of non-commutative cyclic structure. Since we work over an arbitrary field, our techniques may be applied both to l...

Cyclic structures on convolutional codes are modeled using an Ore extension of a finite semisimple algebra A over a finite field . In this context, the separability of the ring extension implies that every ideal code is a split ideal code. We characterize this separability by means of σ being a separable automorphism of the –algebra A. We design an...

We propose a decoding algorithm for a class of convolutional codes called skew Reed-Solomon convolutional codes. These are convolutional codes of designed Hamming distance endowed with a cyclic structure yielding a left ideal of a non-commutative ring (a quotient of a skew polynomial ring). In this setting, right and left division algorithms exist,...

In this paper, we propose a new way of providing cyclic structures to convolutional codes. We define the skew cyclic convolutional codes as left ideals of a quotient ring of a suitable non-commutative polynomial ring. In contrast to the previous approaches to cyclicity for convolutional codes, we use Ore polynomials with coefficients in a field (th...

Let Mn(����) be the algebra of n _ n matrices over the _nite _eld ����. In this paper we prove that the dual code of each ideal convolutional code in the skew-polynomial ring Mn(����)[z;σU] which is a direct summand as a left ideal, is also an ideal convolutional code over Mn(����)[z;σUT] and a direct summand as a left ideal. Moreover we provide an...

In this paper we deal with the theory of rough ideals started in B. Davvaz, Roughness in rings, Information Sciences 164 (1–4) (2004) 147–163. We show that the approximation spaces built from an equivalence relation compatible with the ring structure, i.e. associated to a two-sided ideal, are too naive in order to develop practical applications. We...

Let R be a non-commutative PID finitely generated as a module over its center C. In this paper we give a criterion to decide effectively whether two given elements f,g∈R are similar, that is, if there exists an isomorphism of left R-modules between R/Rf and R/Rg. Since these modules are of finite length, we also consider the more general problem of...

Let (F ⊆ K) an extension of finite fields and (A = Mn K) be the ring of square matrices of order n over (K) viewed as an algebra over (F). Given an (F)--automorphism (σ) on (A) the Ore extension (A[z;σ]) may be used to built certain convolutional codes, namely, the ideal codes. We provide an algorithm to decide if the automorphism (σ) on (A) is a s...

We show that the effective factorization of Ore polynomials over $\mathbb{F}_q(t)$ is still an open problem. This is so because the known algorithm in [1] presents two gaps, and therefore it does not cover all the examples. We amend one of the gaps, and we discuss what kind of partial factorizations can be then computed by using [1].

We show that, under mild conditions of separability, an ideal code, as defined in Lopez-Permouth and Szabo (J Pure Appl Algebra 217(5):958–972, 2013), is a direct summand of an Ore extension and, consequently, it is generated by an idempotent element. We also design an algorithm for computing one of these idempotents.

This paper investigates the application of the theoretical algebraic notion of a separable ring extension, in the realm of cyclic convolutional codes or, more generally, ideal codes. We work under very mild conditions, that cover all previously known as well as new non trivial examples. It is proved that ideal codes are direct summands as left idea...

Let R be an Ore extension of a skew-field. A basic computational problem is to decide effectively whether two given Ore polynomials f, g ∈ R (of the same degree) are similar, that is, if there exists an isomorphism of left R--modules between R/Rf and R/Rg. Since these modules are of finite length, we consider the more general problem of deciding wh...

We develop a fast algorithm for computing the bound of an Ore polynomial over a skew field, under mild conditions. As an application, we state a criterion for deciding whether a bounded Ore polynomial is irreducible, and we discuss a factorization algorithm. The asymptotic time complexity in the degree of the given Ore polynomial is studied. In the...

In this paper, a refinement of the weight distribution in an MDS code is computed. Concretely, the number of codewords with a fixed amount of nonzero bits in both information and redundancy parts is obtained. This refinement improves the theoretical approximation of the information-bit and -symbol error rate, in terms of the channel bit-error rate,...

In this paper we introduce prime fuzzy ideals over a noncommutative ring. This notion of primeness is equivalent to level cuts being crisp prime ideals. It also generalizes the one provided by Kumbhojkar and Bapat in [Not-so-fuzzy fuzzy ideals, Fuzzy Sets and Systems 37 (1990), 237--243], which lacks this equivalence in a noncommutative setting. Se...

We characterize the existence of elimination orderings for a given PBW algebra. Elimination orderings on Np are analyzed. A subclass of elimination orderings is considered to handle some Ore subsets and classical localizations.

The filtered -algebras with a graded AS polynomial ring as associated graded algebra are the AS polynomial rings. In this situation the filtration can be assumed to be finite dimensional. This finiteness allows some applications concerning Gelfand–Kirillov dimension.

While semisimple artinian rings and semisimple coalgebras over a field can be described in terms of matrices (either matrix ring over division rings or comatrix coalgebras over the ground field), semisimple corings seem to have a more intrincated structure in general. It turns out that some well-known properties of semisimple rings or coalgebras, w...

We prove that any multi-filtered algebra with semi-commutative associated graded algebra can be endowed with a locally finite filtration keeping up the semi-commutativity of the associated graded algebra. As consequences, we obtain that Gelfand–Kirillov dimension is exact for finitely generated modules and that the algebra is finitely partitive. Ou...

In this paper the Poincar–Birkhoff–Witt (PBW) rings are characterized. Grbner bases techniques are also developed for these rings. An explicit presentation of Ext i (M,N) is provided when N is a centralizing bimodule.

In this paper we use refiltering methods to prove that certain types of multi-filtered algebras are Auslander-regular and Cohen-Macaulay. This is applied to obtain that the quantized enveloping algebra associated to a Cartan matrix is Auslander-regular and Cohen-Macaulay.

this paper we generalize this test for completely prime ideals in a wide class of iterated Ore extensions of a field. There are interesting examples of algebras where all prime ideals are completely prime. The classical example, the universal enveloping algebra of a finite dimensional solvable Lie algebra over a field of characteristic zero, is sta...

Let $A$ be an algebra over a commutative ring $R$. If $R$ is noetherian and $A^\circ$ is pure in $R^A$, then the categories of rational left $A$-modules and right $A^\circ$-comodules are isomorphic. In the Hopf algebra case, we can also strengthen the Blattner-Montgomery duality theorem. Finally, we give sufficient conditions to get the purity of $...

An abstract is not available.

In this paper we prove a generalization of the primality test in [P. Gianni, B. M. Trager and G. Zacharias, J. Symbolic Comput. 6 (1988), no. 2-3, 149–167], for a wide class of iterated Ore extensions of a commutative polynomial ring.

In this paper we prove a generalization of the [4] primality test for iterated differential operator rings and for coordinate rings of quantum spaces. In fact we prove even more. Our general procedure also applies for some quantum groups like O q (M n (|)), O q (GL n (|)), O q (SL n (|)) for q ∈ | not a root of unity and U + q (g) where g is a semi...

Introduction There is a hard problem in the theory of the noetherian rings, the computation of Gelfand--Kirillov dimension of modules. This problem has been solved in [BCJ96] for universal enveloping algebras of finite--dimensional Lie algebras. In this note we provide a effective method based in the computation of a Grobner basis of a (left, right...

This paper deals with the application of methods stemming from computational commutative algebra to some quantum groups (Weyl algebras, enveloping algebra of a finite dimensional Lie algebra...), using the fact that most quantum groups have PBW-bases

A study of finiteness in a kind of finitely presented quotient algebras is dis-played in this paper. The relations generating the ideals are given by monomials or bi-nomials of same length, in order to obtain homogeneous computations. These ideals are parametrized by 3-tuples (a, b, c), being a the number of variables, b the length of mono-mials an...

We characterize elimination ordering fulfilling that the elements of a given subset of ℤ n are greater than 0. We use this characterization to know if there exists one such ordering and in that case to obtain one. We apply these results to improve some methods in presentations of finitely generated commutative monoids and for computing subalgebras...