Gabriel Navarro

• PhD
• Professor at University of Granada

In recent years, advancements in brain science and neuroscience have significantly influenced the field of computer science, particularly in the domain of reinforcement learning (RL). Drawing insights from neurobiology and neuropsychology, researchers have leveraged these findings to develop novel mechanisms for understanding intelligent decision-m...

In recent years, advancements in brain science and neuroscience have significantly influenced the field of computer science, particularly in the domain of reinforcement learning (RL). Drawing insights from neurobiology and neuropsychology, researchers have leveraged these findings to develop novel mechanisms for understanding intelligent decision-m...

We analyze the connection between two perspectives when defining fuzzy sets: the viewpoint of mappings and the viewpoint of families of level cuts. This analysis is mathematically supported by the framework of a categorical adjunction, which serves as a dictionary between these two perspectives. We prove that hesitant fuzzy sets and gradual sets ar...

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.

Quantum neural networks constitute one of the most promising applications of Quantum Machine Learning, as they leverage both the capabilities of classical neural networks and the unique advantages of quantum mechanics. Moreover, quantum mechanics has demonstrated its ability to detect atypical patterns in data that are challenging for classical app...

In the last few years, deep reinforcement learning has been proposed as a method to perform online learning in energy-efficiency scenarios such as HVAC control, electric car energy management, or building energy management, just to mention a few. On the other hand, quantum machine learning was born during the last decade to extend classic machine l...

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.

In this paper we show a methodology for designing operators on spaces of lattice-valued mappings. More precisely, from a family of operators on a bounded lattice L and mappings from a set X to itself, we may construct an operator, that we call the induced operator, on the lattice of set mappings from X to L. Furthermore, if X is also a bounded latt...

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...

In this paper we deal with the lattice-compatibility between several classes of extended fuzzy sets. Concretely, we treat the problem of finding a lattice structure on set-valued fuzzy sets (SVFSs) whose restriction to interval-valued fuzzy sets (IVFSs) and (type-1) fuzzy sets (FSs) match Zadeh's classical lattice operations. A prominent approach t...

A large class of MDS linear codes is constructed. These codes are endowed with an efficient decoding algorithm. Both the definition of the codes and the design of their decoding algorithm only require from Linear Algebra methods, making them fully accesible for everyone. Thus, the first part of the paper develops a direct presentation of the codes...

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.

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...

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...

A large class of MDS linear codes is constructed. These codes are endowed with an efficient decoding algorithm. Both the definition of the codes and the design of their decoding algorithm only require from Linear Algebra methods, making them fully accesible for everyone. Thus, the first part of the paper develops a direct presentation of the codes...

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...

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....

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...

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...

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...

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,...

We propose a decoding algorithm for a class of convolutional codes called skew BCH 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, so our a...

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 make a first approach to the representation theory of the wedge product of coalgebras by means of the description of its valued Gabriel quiver. Then we define semiprime coalgebras and study its category of comodules by the use of localization techniques. In particular, we prove that, whether its Gabriel quiver is locally finite, any monomial sem...

We introduce the notion of quantum duplicates of an (associative, unital) algebra, motivated by the problem of constructing toy-models for quantizations of certain configuration spaces in quantum mechanics. The proposed (algebraic) model relies on the classification of factorization structures with a two-dimensional factor. In the present paper, ma...

We study serial coalgebras by means of their valued Gabriel quivers. In particular, Hom-computable and representation-directed serial coalgebras are characterized. The Auslander–Reiten quiver of a serial coalgebra is described. Finally, a version of Eisenbud–Griffith Theorem is proved, namely, every subcoalgebra of a prime, hereditary and strictly...

We define the notion of admissible pair for an algebra A, consisting on a couple (Γ, R), where Γ is a quiver and R a unital, splitted and factorizable representation of Γ, and prove that the set of admissible pairs for A is in one to one correspondence with the points of the variety of twisting maps TAn:=T(Kn,A)\mathcal{T}_A^n:=\mathcal{T}(K^n,A)....

We use prime coalgebras as a generalization of simple coalgebras, and observe that prime subcoalgebras represent the structure of the coalgebra in a more efficient way than simple coalgebras. In particular, in this work we focus our attention on the study and characterization of prime subcoalgebras of path coalgebras of quivers and, by extension, o...

We develop the theory of generalized path algebras as defined by Coelho and Xiu [4]. In particular, we focus on the relation between a set of algebras and its associated generalized path algebra for a given quiver. Explicitly, we describe the modules over a generalized path algebra by means of generalized linear rep-resentation of the generalized q...

We apply the theory of localization for tame and wild coalgebras in order to prove the following theorem: “Let Q be an acyclic quiver. Then any tame admissible subcoalgebra of KQ is the path coalgebra of a quiver with relations”.

We study serial coalgebras by means of their valued Gabriel quivers. In particular, Hom-computable and representation-directed coalgebras are characterized. The Auslander-Reiten quiver of a serial coalgebra is described. Finally, a version of Eisenbud-Griffith theorem is proved, namely, every subcoalgebra of a prime, hereditary and strictly quasi-f...

We apply the theory of localization for tame and wild coalgebras in order to prove the following theorem: "Let Q be an acyclic quiver. Then any tame admissible subcoalgebra of KQ is the path coalgebra of a quiver with relations".

We give an explicit description of the set of all factorization structures, or twisting maps, existing between the algebras k^2 and k^2, and classify the resulting algebras up to isomorphism. In the process we relate several different approaches formerly taken to deal with this problem, filling a gap that appeared in a recent paper by Cibils. We al...

We analyze the geometry of the Ext-quiver of a coalgebra $C$ in order to study the behavior of simple and injective $C$-comodules under the action of the functors associated to a localizing subcategory of the category of $C$-comodules.

We study localizing and colocalizing subcategories of a comodule category of a coalgebra C over a field, using the correspondence between local-izing subcategories and equivalence classes of idempotent elements in the dual algebra C * . In this framework, we give a useful description of the localization functor by means of the Morita–Takeuchi conte...

In this paper the notion of path coalgebra of a quiver with relations introduced in (11) and (12) is studied. In particular, developing this topic in the context of the weak* topology, we give a criterion that allows us to verify whether or not a relation subcoalgebra of a path coalgebra is a path coalgebra of a quiver with relations. 1. Introducti...

In this article we review recent developments in representation theory of coalgebras, aiming for an extension of the classical theory for artinian algebras. The key tool is the use of the theory of localization in categories of comodules and, in particular, the behaviour of simple comodules through the action of the section functor. For that reason...

We deal with the problem of describing quantum duplicates of finite set algebras, giving especial attention to the ones obtained using as duplication factor the algebra k( )/( 2) of dual numbers.

In the present paper, we extend the combinatorial techniques used by C. Cibils, defining the notion of admissible pair for an algebra A, consisting on a couple ( , R) where is a quiver, and R a representation of satisfying certain restrictions (namely, to be unital, splitted and factorizable), and prove that the variety of twisting maps TA := T (Kn...