In this work we consider repeated-root multivariable codes over a finite chain ring. We show cond... more In this work we consider repeated-root multivariable codes over a finite chain ring. We show conditions for these codes to be principally generated. We consider a suitable set of generators of the code and compute its minimum distance. As an application we study the relevant example of the generalized Kerdock code in its r-dimensional cyclic version.
The structure of multivariate semisimple codes over a finite chain ring $R$ is established using ... more The structure of multivariate semisimple codes over a finite chain ring $R$ is established using the structure of the residue field $\bar R$. Multivariate codes extend in a natural way the univariate cyclic and negacyclic codes and include some non-trivial codes over $R$. The structure of the dual codes in the semisimple abelian case is also derived and some conditions on the existence of selfdual codes over $R$ are studied.
A Generalized Galois Ring (GGR) S is a finite nonassociative ring with identity of characteristic... more A Generalized Galois Ring (GGR) S is a finite nonassociative ring with identity of characteristic p n , for a prime number p, such that its top-factor [`</font >(S)] = S/pS\overline{S} = S/pS is a finite semifield. It is well known that if S is an associative Galois Ring (GR) then the set S* = S pSS^* = S \
Proceedings of the 2009 international symposium on Symbolic and algebraic computation - ISSAC '09, 2009
... Irene Polo-Blanco Universidad de Cantabria Departamento de Matemáticas, Estadística y Computa... more ... Irene Polo-Blanco Universidad de Cantabria Departamento de Matemáticas, Estadística y Computación 39005 Santander, Spain irene.polo@unican.es Jorge Caravantes Università degli Studi di Genova Dipartimento di Matematica 16146 Genova, Italy caravant@dima.unige.it ...
ABSTRACT The structure of multivariate serial codes over a finite chain ring R is established usi... more ABSTRACT The structure of multivariate serial codes over a finite chain ring R is established using the structure of the residue field R ¯. Multivariate codes extend in a natural way the univariate cyclic and negacyclic codes and include some nontrivial codes over R. The structure of the dual codes in the serial abelian case is also derived, and some conditions for the existence of self-dual codes over R are studied.
A finite semifield D is a finite nonassociative ring with identity such that the set D * =D set m... more A finite semifield D is a finite nonassociative ring with identity such that the set D * =D set minus {0} is closed under the product. In this paper we obtain a computer-assisted description of all semifields of order 64, which completes the classification of finite semifields of order at ...
International Journal of Computer Mathematics, 2011
Finite semifields (finite non-necessarily associative division rings) have traditionally been con... more Finite semifields (finite non-necessarily associative division rings) have traditionally been considered in the context of finite geometries (they coordinatize projective semifield planes). New applications to the coding theory, combinatorics and the graph theory have broadened the potential interest in these rings. We show recent progress in the study of these objects with the help of computational tools. In particular, we
International Journal of Computer Mathematics, 2012
In this paper, a highly demanding computer-assisted classification of four-dimensional finite sem... more In this paper, a highly demanding computer-assisted classification of four-dimensional finite semifields over the field 𝔽7 is provided. The techniques considered to overcome the difficulties in the management of the large data processed are explained.
International Journal of Algebra and Computation, 2007
ABSTRACT A finite semifield D is a finite nonassociative ring with identity such that the set D* ... more ABSTRACT A finite semifield D is a finite nonassociative ring with identity such that the set D* = D \{0} is a loop under the product. Wene conjectured in [1] that any finite semifield is either right or left primitive, i.e. D* is the set of right (or left) principal powers of an element in D. In this paper we study the primitivity of finite semifields with 64 and 81 elements.
ABSTRACT Additive multivariate codes over F4F4 (the Galois field with 4 elements) are a natural e... more ABSTRACT Additive multivariate codes over F4F4 (the Galois field with 4 elements) are a natural extension of additive cyclic and abelian codes. A complete description of such codes when the length is odd was presented in [11]. In this paper we study some properties of this family of codes in the case when the length is even (modular case) and the number of variables is two (bivariate codes).
In this work we consider repeated-root multivariable codes over a finite chain ring. We show cond... more In this work we consider repeated-root multivariable codes over a finite chain ring. We show conditions for these codes to be principally generated. We consider a suitable set of generators of the code and compute its minimum distance. As an application we study the relevant example of the generalized Kerdock code in its r-dimensional cyclic version.
The structure of multivariate semisimple codes over a finite chain ring $R$ is established using ... more The structure of multivariate semisimple codes over a finite chain ring $R$ is established using the structure of the residue field $\bar R$. Multivariate codes extend in a natural way the univariate cyclic and negacyclic codes and include some non-trivial codes over $R$. The structure of the dual codes in the semisimple abelian case is also derived and some conditions on the existence of selfdual codes over $R$ are studied.
A Generalized Galois Ring (GGR) S is a finite nonassociative ring with identity of characteristic... more A Generalized Galois Ring (GGR) S is a finite nonassociative ring with identity of characteristic p n , for a prime number p, such that its top-factor [`</font >(S)] = S/pS\overline{S} = S/pS is a finite semifield. It is well known that if S is an associative Galois Ring (GR) then the set S* = S pSS^* = S \
Proceedings of the 2009 international symposium on Symbolic and algebraic computation - ISSAC '09, 2009
... Irene Polo-Blanco Universidad de Cantabria Departamento de Matemáticas, Estadística y Computa... more ... Irene Polo-Blanco Universidad de Cantabria Departamento de Matemáticas, Estadística y Computación 39005 Santander, Spain irene.polo@unican.es Jorge Caravantes Università degli Studi di Genova Dipartimento di Matematica 16146 Genova, Italy caravant@dima.unige.it ...
ABSTRACT The structure of multivariate serial codes over a finite chain ring R is established usi... more ABSTRACT The structure of multivariate serial codes over a finite chain ring R is established using the structure of the residue field R ¯. Multivariate codes extend in a natural way the univariate cyclic and negacyclic codes and include some nontrivial codes over R. The structure of the dual codes in the serial abelian case is also derived, and some conditions for the existence of self-dual codes over R are studied.
A finite semifield D is a finite nonassociative ring with identity such that the set D * =D set m... more A finite semifield D is a finite nonassociative ring with identity such that the set D * =D set minus {0} is closed under the product. In this paper we obtain a computer-assisted description of all semifields of order 64, which completes the classification of finite semifields of order at ...
International Journal of Computer Mathematics, 2011
Finite semifields (finite non-necessarily associative division rings) have traditionally been con... more Finite semifields (finite non-necessarily associative division rings) have traditionally been considered in the context of finite geometries (they coordinatize projective semifield planes). New applications to the coding theory, combinatorics and the graph theory have broadened the potential interest in these rings. We show recent progress in the study of these objects with the help of computational tools. In particular, we
International Journal of Computer Mathematics, 2012
In this paper, a highly demanding computer-assisted classification of four-dimensional finite sem... more In this paper, a highly demanding computer-assisted classification of four-dimensional finite semifields over the field 𝔽7 is provided. The techniques considered to overcome the difficulties in the management of the large data processed are explained.
International Journal of Algebra and Computation, 2007
ABSTRACT A finite semifield D is a finite nonassociative ring with identity such that the set D* ... more ABSTRACT A finite semifield D is a finite nonassociative ring with identity such that the set D* = D \{0} is a loop under the product. Wene conjectured in [1] that any finite semifield is either right or left primitive, i.e. D* is the set of right (or left) principal powers of an element in D. In this paper we study the primitivity of finite semifields with 64 and 81 elements.
ABSTRACT Additive multivariate codes over F4F4 (the Galois field with 4 elements) are a natural e... more ABSTRACT Additive multivariate codes over F4F4 (the Galois field with 4 elements) are a natural extension of additive cyclic and abelian codes. A complete description of such codes when the length is odd was presented in [11]. In this paper we study some properties of this family of codes in the case when the length is even (modular case) and the number of variables is two (bivariate codes).
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Papers by Ignacio Rua