José Gómez-Torrecillas

• PhD
• Professor at University of Granada

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.

Roos’ direct sum decomposition of a cyclic convolutional code based upon a suitable construction of a minimal encoder is extended to a broader class of convolutional codes endowed with additional algebraic structure. The two (equivalent) usual concepts of a convolutional code are taken into account, as a vector space and as a module.

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.

Asteroseismology, that is, the use of the frequency content of a time series caused by variations in brightness or radial velocity of a stellar object, is based on the hypothesis that such a series is harmonic and therefore can be described by a sum of sines and cosines. If this were not the case (e.g., the oscillations of an ellipsoid of revolutio...

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...

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...

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....

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 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...

Based on the novel notion of `weakly counital fusion morphism', regular weak multiplier bimonoids in braided monoidal categories are introduced. They generalize weak multiplier bialgebras over fields and multiplier bimonoids in braided monoidal categories. Under some assumptions the so-called base object of a regular weak multiplier bimonoid is sho...

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...

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 prove that the classical result asserting that the relative Picard group of a faithfully flat extension of commutative rings is isomorphic to the first Amitsur cohomology group stills valid in the realm of symmetric monoidal categories. To this end, we built some group exact sequences from an adjunction in a bicategory, which are of independent...

We prove that the classical result asserting that the relative Picard group of a faithfully flat extension of commutative rings is isomorphic to the first Amitsur cohomology group stills valid in the realm of symmetric monoidal categories. To this end, we built some group exact sequences from an adjunction in a bicategory, which are of independent...

We construct a contravariant functor (\emph{the finite dual functor}) from the category of co-commutative Hopf algebroids to the category of commutative Hopf algebroids. Using this functor, we then show that the representation theory of a given Lie-Rinehart algebra $(A,L)$ with $A$ is a Dedekind domain, is the 'same' as the representation theory of...

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...

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 recognise Harada's generalized categories of diagrams as a particular case of modules over a monad defined on a finite direct product of additive categories. We work in the dual (albeit formally equivalent) situation, that is, with comodules over comonads. With this conceptual tool at hand, we obtain several of the Harada results with simpler pr...

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...

The present text surveys some relevant situations and results where basic Module Theory interacts with computational aspects of operator algebras. We tried to keep a balance between constructive and algebraic aspects.

Quantum bounded extensions of multifiltered algebras are defined, and they are endowed with a new multifiltration. The behavior of the associated multigraded rings is studied. A refiltering process allows to lift homological properties from the associated multigraded ring to the multifiltered ring. This is the full version, with proofs, of the pape...

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...

Weak (Hopf) bialgebras are described as (Hopf) bimonoids in appropriate duoidal (also known as 2-monoidal) categories. This interpretation is used to define a category wba of weak bialgebras over a given field. As an application, the "free vector space" functor from the category of small categories with finitely many objects to wba is shown to poss...

A non-unital generalization of weak bialgebra is proposed with a multiplier-valued comultiplication. Certain canonical subalgebras of the multiplier algebra (named the `base algebras') are shown to carry coseparable co-Frobenius coalgebra structures. Appropriate modules over a weak multiplier bialgebra are shown to constitute a monoidal category vi...

We show that a left CF ring is left artinian if and only if it is von Neumann regular modulo its left singular ideal. We deduce that a left FGF is Quasi-Frobenius (QF) under this assumption. This clarifies the role played by the Jacobson radical and the singular left ideal in the FGF and CF conjectures. In Sec. 3 of the paper, we study the structur...

Firm Frobenius algebras are firm algebras and counital coalgebras such that the comultiplication is a bimodule map. They are investigated by categorical methods based on a study of adjunctions and lifted functors. Their categories of comodules and of firm modules are shown to be isomorphic if and only if a canonical comparison functor from the cate...

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,...

Given a weak distributive law between algebras underlying two weak bialgebras, we present sufficient conditions under which the corresponding weak wreath product algebra becomes a weak bialgebra with respect to the tensor product coalgebra structure. When the weak bialgebras are weak Hopf algebras, then the same conditions are shown to imply that t...

We show that the functor from bialgebras to vector spaces sending a bialgebra to its subspace of primitives has monadic length at most 2.

We give a bicategorical version of the main result of Masuoka (Tsukuba J Math 13:353–362, 1989) which proposes a non-commutative version of the fact that for a faithfully flat extension of commutative rings , the relative Picard group Pic(S/R) is isomorphic to the Amitsur 1–cohomology group H 1(S/R,U) with coefficients in the units functor U.

We give a bicategorical version of the main result of A. Masuoka ({Corings and invertible bimodules,} {\em Tsukuba J. Math.} \textbf{13} (1989), 353--362) which proposes a non-commutative version of the fact that for a faithfully flat extension of commutative rings $R \subseteq S$, the relative Picard group $Pic(S/R)$ is isomorphic to the Amitsur 1...

Let $R$ be a ring with a set of local units, and a homomorphism of groups $\underline{\Theta} : \G \to \Picar{R}$ to the Picard group of $R$. We study under which conditions $\underline{\Theta}$ is determined by a factor map, and, henceforth, it defines a generalized crossed product with a same set of local units. Given a ring extension $R \subsete...

We study the (so-called bilinear) factorization problem answered by a weak wreath product (of monads and, more specifically, of algebras over a commutative ring) in the works by Street and by Caenepeel and De Groot. A bilinear factorization of a monad R turns out to be given by monad morphisms A --> R <-- B inducing a split epimorphism of B-A bimod...

Let R be a ring with a set of local units, and a homomorphism of groups (Theta) under bar : G -> Pic(R) to the Picard group of R. We study under which conditions (Theta) under bar is determined by a factor map, and, henceforth, it defines a generalized crossed product with a same set of local units. Given a ring extension R subset of S with the sam...

We give a characterization, in terms of Galois infinite comatrix corings, of the corings that decompose as a direct sum of left comodules which are finitely generated as left modules. Then we show that the associated rational functor is exact. This is the case of a right semiperfect coring which is locally projective and whose Galois comodule is a...

Consider a coring with exact rational functor, and a finitely generated and projective right comodule. We construct a functor (\emph{coinduction functor}) which is right adjoint to the hom-functor represented by this comodule. Using the coinduction functor, we establish a bijective map between the set of representative classes of torsion simple rig...

We focus our attention to the set Gr(ℭ) of grouplike elements of a coring ℭ over a ring A. We do some observations on the actions of the groups U(A) and Aut(ℭ) of units of A and of automorphisms of corings of ℭ, respectively, on Gr(ℭ), and on the subset Gal(ℭ) of all Galois grouplike elements. Among them, we give conditions on ℭ under which Gal(ℭ)...

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 develop a Galois (descent) theory for comonads within the framework of bicategories. We give generalizations of Beck's theorem and the Joyal-Tierney theorem. Many examples are provided, including classical descent theory, Hopf-Galois theory over Hopf algebras and Hopf algebroids, Galois theory for corings and group-corings, and Morita-Takeuchi t...

We describe how some aspects of abstract localization on module categories have applications to the study of injective comodules over some special types of corings. We specialize the general results to the case of Doi-Koppinen modules, generalizing previous results in this setting.

The Jacobson–Bourbaki Theorem for division rings was formulated in terms of corings by Sweedler in [M.E. Sweedler, The predual theorem to the Jacobson–Bourbaki Theorem, Trans. Amer. Math. Soc. 213 (1975) 391-406]. Finiteness conditions hypotheses are not required in this new approach. In this paper we extend Sweedler's result to simple artinian rin...

A Bernstein-type inequality is found for a simple localization of the coordinate ring of the quantized symplectic space 𝒪q(𝔰𝔭 ), where q is not a root of unity. Some holonomic modules and an analog of the Bernstein polynomial are also computed.On trouve une inégalité de type “inégalité de Bernstein” pour une localisation simple de l'anneaux des coo...

We investigate which aspects of recent developments on Galois corings and comodules admit a formulation in terms of comonads. This approach hopefully will permit of focusing in what is specific in each particular future situation, having some relevant general results for granted. The general theory is applied to the study of comodules over corings...

Starting from a comonad G on a category A, and a functor L : B -> A with a right adjoint R : A -> B, we will give a parametrization of the functors K from B to the category of all G-coalgebras that factorize throughout L in terms of homomorphisms of comonads from LR to G. Next, we will see under which conditions one of these functors K admits a rig...

We give a notion of a comatrix coring which embodies all former constructions and, what is more interesting, leads to the formulation of a notion of Galois coring and the statement of a Faithfully Flat Descent Theorem that generalize the previous versions.

The Jacobson-Bourbaki Theorem for division rings was formulated in terms of corings by Sweedler in 1975. Finiteness conditions hypotheses are not required in this new approach. In this paper we extend Sweedler's result to simple artinian rings using a particular class of corings, comatrix corings. A Jacobson-Bourbaki like correspondence for simple...

We extend Masuoka's Theorem [11] concerning the isomorphism between the group of invertible bimodules in a non-commutative ring extension and the group of automorphisms of the associated Sweedler's canonical coring, to the class of finite comatrix corings introduced in [6].

We investigate Frobenius pairs between categories of comodules over rather general corings. We particularize to the case of the adjoint pair of functors associated to a morphism of corings over different base rings, which leads to a reasonable notion of Frobenius coring extension. When applied to corings stemming from entwining structures, we obtai...

To a B-coring and a (B,A)-bimodule that is finitely generated and projective as a right A-module an A-coring is associated. This new coring is termed a base ring extension of a coring by a module. We study how the properties of a bimodule such as separability and the Frobenius properties are reflected in the induced base ring extension coring. Any...

To a B-coring and a (B,A)-bimodule that is finitely generated and projective as a right A-module an A-coring is associated. This new coring is termed a base ring extension of a coring by a module. We study how the properties of a bimodule such as separability and the Frobenius properties are reflected in the induced base ring extension coring. Any...

In this paper we extend the theory of serial and uniserial finite dimensional algebras to coalgebras of arbitrary dimension. Nakayama–Skorniakov Theorems are proved in this new setting and the structure of such coalgebras is determined up to Morita–Takeuchi equivalences. Our main structure theorem asserts that over an algebraically closed field k t...

We introduce the notion of right strictly quasi-finite coalgebras, as coalgebras with the property that the class of quasi-finite right comodules is closed under factor comodules, and study its properties. A major tool in this study is the local techniques, in the sense of abstract localization.

We characterize the corings whose category of comodules has a generating set of small projective comodules in terms of the (noncommutative) descent theory. In order to extricate the structure of these corings, we give a generalization of the notions of comatrix coring and Galois comodule which avoid finiteness conditions. A sufficient condition for...

Dans ce travail, on démontre que les cliques d'idéaux premiers d'une classe d'extensions de Ore itérées coı̈ncident avec leurs orbites par rapport à une opération d'un groupe abélien bien déterminé. Cette classe est une sous-classe de celle étudiée dans (S.-Q. Oh, Comm. Algebra 25 (1) (1997) 37–49) et elle contient l'algèbre de Weyl quantique , les...

For any left R-module P with endomorphism ring S, the adjoint pair of functors P⊗S− and HomR(P,−) induce an equivalence between the categories of P-static R-modules and P-adstatic S-modules. In particular, this setting subsumes the Morita theory of equivalences between module categories and the theory of tilting modules. In this paper we consider,...

To any bimodule that is finitely generated and projective on one side one can associate a coring, known as a comatrix coring. A new description of comatrix corings in terms of data reminiscent of a Morita context is given. It is also studied how properties of bimodules are reflected in the associated comatrix corings. In particular it is shown that...

In this chapter, we introduce most of the basic objects and tools that will be used throughout the rest of this book. We start with some background on the notion of admissible order ℕn and show how this may be used to define left Poincaré-Birkhoff-Witt (PBW) rings. The elements of these rings exhibit a behaviour which is similar to that of ordinary...

Until now, we have only dealt with ideals of PBW rings and shown how Gröbner bases may efficiently be used to solve classical questions like the membership or equality problem. In the present chapter we will generalize this approach and show how it applies as well to submodules of the free module R n Actually, it appears that this set-up may also b...

We start this chapter with some generalities concerning classsical localication at Ore sets. Very few proofs are included, as this topic is well documented in the literature, cf. [61, 91, 116], for example.

This chapter starts with some first “classical” applications, such as the solution of the membership or equality problem for ideals in a PBW ring. Since many of these applications are non-commutative analogs of corresponding ones in the commutative case, our presentation runs along the lines of that in “classical” texts, like [2, 11, 41], for examp...

In contrast with the commutative case, for non-commutative algebras the classical Krull dimension is usually not a very useful tool. Indeed, as this notion is defined by using chains of (two-sided) prime ideals, it only makes sense for rings having “many” two-sided ideals. Fortunately, for finitely generated k-algebras R, we may define the so-calle...

A PBW algebra R over a field k may be viewed as an associative algebra generated by finitely many elements x 1,…, x n subject to the relations $$Q = \{ {x_j}{x_i} = {q_{ji}}{x_i}{x_j} + {p_{ji}}\} \;(1 \leqslant i < j \leqslant n)$$, where each p ji is a finite k-linear combination of standard terms \({x^\alpha } = x_1^{{\alpha _1}} \cdots x_n^{{\a...

The main purpose of this introductory chapter is to provide a short survey of some background on rings and modules, which will be used throughout this book. We did, of course, not aim at a complete or self-contained exposition, as this would fall outside of the framework of the present text. Instead we restrict ourselves exactly to those topics, de...

As we pointed out before, several problems within ideal and module theory, such as the computation of intersections, may be solved by using elimination techniques. However, it appears that this approach is highly inefficient from the computational point of view, mainly due to the necessary use of the lex ordering (or, more generally, any eliminatio...

In order to extrincate the structure of corings with a finitely generated and projective generator we give the notion of a comatrix coring. As consequences we give generalizations of the main characterizations of faithfully flat Galois corings and extensions which work for corings without grouplike elements, as well as a generalization of the Desce...

We investigate the relationship between coseparable and semisimple corings. In particular we prove that a coring over a separable algebra is coseparable if and only if it is absolutely semisimple.

We offer an approach to basic coalgebras with inspiration in the classical theory of idempotents for finite dimensional algebras. Our theory is based upon the fact that the co-hom functors associated to direct summands of the coalgebra can be easily described in terms of idempotents of the convolution algebra. Our approach is shown to be equivalent...

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...

The group of automorphisms of the coordinate ring of quantum sym- plectic space Oq(spk2◊n) is isomorphic to the algebraic torus (k◊)n+1, when q is not a root of unity.

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.