Publicación: Construcción de curvas algebraicas maximales sobre cuerpos finitos
Cargando...
Fecha
2018-10-22
Autores
Editor/a
Director/a
Tutor/a
Coordinador/a
Prologuista
Revisor/a
Ilustrador/a
Derechos de acceso
Atribución-NoComercial-SinDerivadas 4.0 Internacional
info:eu-repo/semantics/openAccess
info:eu-repo/semantics/openAccess
Título de la revista
ISSN de la revista
Título del volumen
Editor
Universidad Nacional de Educación a Distancia (España). Facultad de Ciencias
Resumen
Este trabajo describe la construcción de curvas maximales sobre cuerpos finitos, cubiertas por curvas hermitianas. Existe un gran interés en las curvas algebraicas sobre cuerpos finitos con muchos puntos racionales, ya que son utilizadas para la construcción de códigos correctores de errores, es decir, añadir información redundante a un mensaje con el propósito de ser recuperado en el caso de producirse errores . Explicamos las herramientas necesarias para abordar este tema, como parte de la Teoría de Anillos y Cuerpos, y analizamos el artículo On Certain Subcovers of the hermitian Curve, escrito por Arnaldo Garcá, Motoko Q. Kawakita y Shinji M, donde se detalla un método para la construcción de estas curvas maximales. Al final del trabajo, se presentan varias tablas describiendo diferentes representaciones de algunos cuerpos finitos y las tablas de los puntos racionales de las curvas maximales que hemos obtenido aplicando el método descrito en el artículo. Todas estas tablas han sido obtenidas mediante la ejecución de un programa desarrollado en JAVA, ad-hoc, el cual puede ser generalizado para cualquier cuerpo finito y la evaluación de sus puntos en cualquier curva dada.
This thesis surveys the construction of maximal curves over finite fields covered by Hermitian curves. Currently there is a great interest in algebraic curves over finite field which have many rational points, because they are widely used for the construction of error correcting codes, that means, adding redundant data to a message in order to be recovered even when errors are introduced. We explain the necessary tools to address this topic, like part of the Ring and Field Theory and we analyze the paper titled On Certain Subcovers of the Hermitian Curve, written by Arnaldo García, Motoko Q. Kawakita and Shinji M, where it is explained in detail one method to construct such that maximal curves. At the end of this thesis, we present some tables describing different representations of finite fields, and some tables containing the rational points of the maximal curves obtained by using the method described in that paper. All these tables have been generated through the execution of one application developed in JAVA programming language, ad-hoc, which could be generalized to get any finite field and for the evaluation of its points in any given curve.
This thesis surveys the construction of maximal curves over finite fields covered by Hermitian curves. Currently there is a great interest in algebraic curves over finite field which have many rational points, because they are widely used for the construction of error correcting codes, that means, adding redundant data to a message in order to be recovered even when errors are introduced. We explain the necessary tools to address this topic, like part of the Ring and Field Theory and we analyze the paper titled On Certain Subcovers of the Hermitian Curve, written by Arnaldo García, Motoko Q. Kawakita and Shinji M, where it is explained in detail one method to construct such that maximal curves. At the end of this thesis, we present some tables describing different representations of finite fields, and some tables containing the rational points of the maximal curves obtained by using the method described in that paper. All these tables have been generated through the execution of one application developed in JAVA programming language, ad-hoc, which could be generalized to get any finite field and for the evaluation of its points in any given curve.
Descripción
Categorías UNESCO
Palabras clave
group, ideal, field, Nullstellensatz, variety, rational function field, curve, field extensions, Riemann-Roch theorem, ramification, covering, Hasse-Weils bound
Citación
Centro
Facultad de Ciencias
Departamento
Matemáticas Fundamentales