Ignacio F. Rúa

Determination of division algebras with 243 elements

Finite nonassociative division algebras (i.e., finite semifields) with 243 elements are completely classified.

On repeated-root multivariable codes over a finite chain ring

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.

New advances in the computational exploration of semifields

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 state results on the classification and primitivity of semifields obtained with the help of advanced and efficient implementations (both sequential and parallel) of different algorithms specially designed to manipulate these objects.

Additive semisimple multivariable codes over F4

The structure of additive multivariable codes over

Solving the implicitization, inversion and reparametrization problems for rational curves through subresultants

{\mathbb{F}_4}

Analyzing group based matrix multiplication algorithms

(the Galois field with 4 elements) is presented. The semisimple case (i.e., when the defining polynomials of the code have no repeated roots) is specifically addressed. These codes extend in a natural way the abelian codes, of which additive cyclic codes of odd length are a particular case. Duality of these codes is also studied.

Computing the topology of a real algebraic plane curve whose equation is not directly available

Abstract Codes over finite commutative chain rings have been introduced as a generalization of codes over finite fields. Let S | R be a Galois extension of finite commutative chain rings. If C ⊆ S n is an S-code, it is possible to define, starting from C, two different R-codes: Res ( C ) = C ∩ R n and Tr ( C ) , where Tr is the trace function. In this work we analyze the relationships between these R-codes and the duality operator.

On Cyclic Top-Associative Generalized Galois Rings

From the rational functions defining a rational plane curve it is possible to construct two bivariate polynomials that can be seen as univariate polynomials in the parameter value. In this paper several relevant properties of the subresultant sequence of these two polynomials are introduced which are used to solve simultaneously the implicitization, inversion, properness and reparametrization problems. It is also shown that these methods can be suitably transformed in order to deal with curves in the three-dimensional space presented by a polynomial parametrization.