José María Muñoz Porras

Universidad de Salamanca · Department of Mathematics

The performance of Rosenthal’s decoding algorithm is measured when it is applied to certain family of 1-dimensional MDS convolutional codes. It is shown that the performance of this algorithm depends on the algebraic structure of the convolutional code we are working with.

Convolutional Goppa codes (CGC) were defined in Appl. Algebra Eng. Comm. Comput., vol. 15, pp. 51-61, 2004 and IEEE Trans. Inf. Theory, vol. 52, 340-344, 2006. In this paper, we prove that every convolutional code is a CGC defined over a smooth curve over BBF q(z) and we give an explicit construction of convolutional codes as CGC over the projectiv...

In this article we study coverings with prescribed ramification from the point of view of the Sato Grassmannian and of the algebro-geometric theory of solitons. We show that the moduli space of such coverings, which is a Hurwitz scheme, is a subscheme of the Grassmannian. We give its equations and show that there is a Virasoro group that uniformize...

In this paper we show the existence of a group acting infinitesimally transitively on the moduli space of pointed-curves and vector bundles (with formal trivialization data) and whose Lie algebra is an algebra of differential operators. The central extension of this Lie algebra induced by the determinant bundle on the Sato Grassmannian is precisely...

We give a general method to construct MDS one-dimensional convolutional codes. Our method generalizes previous constructions of H. Gluesing-Luerssen and B. Langfeld. Moreover we give a classification of one-dimensional Convolutional Goppa Codes and propose a characterization of MDS codes of this type.

Se hará una introducción a la teoría de códigos convolucionales y a continuación describiremos la construcción de códigos convolucionales como códigos de valoración en una familia uniparamétrica de curvas algebraicas. Los casos de las curvas de géneros cero y uno se estudiaran con detalle.

We define a new class of Convolutional Codes in terms of fibrations of algebraic varieties generalizaing our previous constructions of Convolutional Goppa Codes. Using this general construction we can give several examples of Maximum Distance Separable (MDS) Convolutional Codes. Comment: 10 pages

Using the technique of the Fourier-Mukai transform we give an explicit set of generators of the ideal defining an algebraic curve as a subscheme of its Jacobian. Essentially, these ideals are generated by the Fay's trisecant identities.

We obtain a characterization of theta functions of Jacobian varieties of curves with automorphisms among theta functions of principally polarized abelian varieties (p.p.a.v.). We first give a characterization in terms of finite dimensional orbits for a suitable action in the Sato Grassmannian. Secondly, the introduction of formal Baker-Akhiezer fun...

Convolutional codes have the structure of an F[z]-module. To study their properties it is desirable to classify them as the points of a certain algebraic variety. By considering the correspondence of submodules and the points of certain quotient schemes, and the inclusion of these as subvarieties of certain Grassmannians, one has a one-to-one corre...

This paper is devoted to the study of the uniformization of the moduli space of pairs (X, E) consisting of an algebraic curve and a vector bundle on it. For this goal, we study the moduli space of 5-tuples (X, x, z, E, \phi), consisting of a genus g curve, a point on it, a local coordinate, a rank n degree d vector bundle and a formal trivializatio...

There are excellent surveys on the history of the Schottky problem like [16]. Our aim in this paper is to expose the approaches to the Schottky problem more directly related to the recent results proved by I. Krichever and S. Grushevsky ([8, 9, 12, 13]). That is, we will expose the approaches related with the existence of trisecants, the KP equatio...

Las técnicas de geometría algebraica para construir códigos lineales pueden ser aplicados a la construcción de códigos convolucionales, usando curvas algebraicas sobre los campos de función. En este sentido se construyen códigos Goppa convolucionales y se provee un sistema para construir códigos convolucionales con propiedades prescritas. Algebraic...

The main result proved in the paper is the computation of the explicit equations defining the Hurwitz schemes of coverings with punctures as subschemes of the Sato infinite Grassmannian. As an application, we characterize the existence of certain linear series on a smooth curve in terms of soliton equations.

It is shown that there exists a geometric quotient of the subscheme of stable points of under the action of . The consequences in terms of vector bundles on an algebraic curve are studied.

The action of on the Sato Grassmannian is studied. Following ideas similar to those of GIT and to those used in the study of vector bundles, the (semi)stable points are introduced. It is shown that any point admits a Harder–Narasimhan filtration and that, if it is semistable, it has a Jordan–Hölder filtration. Finally, theses results are compared w...

In this paper the moduli space of Higgs pairs over a fixed smooth projective curve with extra formal data is defined and it is endowed with a scheme structure. We introduce a relative version of the Krichever map using a fibration of Sato Grassmannians and show that this map is injective. This fact and the characterization of the points of the imag...

Se definirán las funciones theta de una variable y se explicará su relación con la teoría de funciones elípticas. Se dará una introducción a la teoría de formas modulares y a la relación entre la función theta y la función zeta de Riemann. Finalmente comentaremos las generalizaciones a varias variables y su interpretación geométrica en términos de...

En el artículo se definen los códigos Goppa a través de curvas algebraicas y la construccion de sus correspondientes códigos duales. In this correspondence, we define convolutional Goppa codes over algebraic curves and construct their corresponding dual codes. Examples over the projective line and over elliptic curves are described, obtaining in pa...

In this correspondence, we define convolutional Goppa codes over algebraic curves and construct their corresponding dual codes. Examples over the projective line and over elliptic curves are described, obtaining in particular some maximum-distance separable (MDS) convolutional codes.

In this short note we give a characterization of extremal principally polarized abelian varieties determining an isolated point in Sing The case g = 5 is treated in depth.

In this paper we study the Hurwitz scheme in terms of the Sato Grassmannian and the algebro-geometric theory of solitons. We will give a characterization, its equations and a show that there is a group of Virasoro type which uniformizes it.

In this short note we give a characterization of extremal principally polarized abelian varieties determining an isolated point in $Sing \Cal A_g$. The case $g=5$ is treated with detail.

Se definen nuevos códigos convolucionales (MDS) generalizando los códigos Goppa. A new kind of Convolutional Codes generalizing Goppa Codes is proposed. This provides a systematic method for constructing convolutional codes with prefixed properties. In particular, examples of Maximum-Distance Separable (MDS) convolutional codes are obtained.

The aim of the paper is twofold. First, some results of Shiota and Plaza-Martn on Prym varieties of curves with an involution are generalized to the general case of an arbitrary automorphism of prime order. Second, the equations defining the moduli space of curves with an automorphism of prime order as a subscheme of the Sato Grassmannian are given...

We consider a relative Fourier-Mukai transform defined on elliptic fibrations over an arbitrary normal base scheme. This is used to construct relative Atiyah sheaves and generalize Atiyah's and Tu's results about semistable sheaves over elliptic curves to the case of elliptic fibrations. Moreover we show that this transform preserves relative (semi...

We characterize the subscheme of the moduli space of torsion-free sheaves on an elliptic surface which are stable of relative degree zeero (with respect to a polarization of type aH+bf, H being the section and f the elliptic fibre) which is isomorphic, via the relative Fourier-Mukai transform, with the relative compactified Simpson Jacobian of the...

In our earlier paper it was proved that the singular locus of A g (coarse moduli space of principally polarized abelian varieties over C) is ex-pressed as the union of irreducible varieties A g (p, α) representing abelian va-rieties with an order p automorphism of fixed entire representation. In this paper we prove that A g (p, α) is an irreducible...

This paper is concerned with the formulation of a non-pertubative theory of the bosonic string. We introduce a formal group $G$ which we propose as the ``universal moduli space'' for such a formulation. This is motivated because $G$ establishes a natural link between representations of the Virasoro algebra and the moduli space of curves. Among othe...

We use a relative Fourier–Mukai transform on elliptic K3 surfaces X to describe mirror symmetry. The action of this Fourier–Mukai transform on the cohomology ring of X reproduces relative T-duality and provides an infinitesimal isometry of the moduli space of algebraic structures on X which, in view of the triviality of the quantum cohomology of K3...

The main result of this paper is the explicit computation of the equations defining the moduli space of triples $(C,p,z)$ (where $C$ is an integral and complete algebraic curve, $p$ a smooth rational point and $z$ a formal trivialization around $p$) in the infinite Grassmannian of $k((t))$. This is achieved by introducing infinite Grassmannians, ta...

The aim of this paper is to offer an algebraic construction of infinite-dimensional Grassmannians and determinant bundles (and therefore valid for arbitrary base fields). As an application we construct the $\tau$-function and formal Baker-Akhiezer functions over arbitrary fields, by proving the existence of a ``formal geometry'' of local curves ana...

A proof of the conjecture of Belavin, Knizhnik and Morozov concerning the Polyakov measure of the bosonic string in genus 4 is offered. Its consequences for the universal moduli space approach to string theory are discussed.