Algebraic Curves over Finite Fields

Rovi, Carmen. (2010). Algebraic Curves over Finite Fields Master Thesis, Universidad Nacional de Educación a Distancia (España). Facultad de Ciencias y Linköpings Universitet. MAI Mathematics

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Título Algebraic Curves over Finite Fields
Autor(es) Rovi, Carmen
Abstract This thesis surveys the issue of finding rational points on algebraic curves over finite fields. Since Goppa’s construction of algebraic geometric codes, there has been great interest in finding curves with many rational points. Here we explain the main tools for finding rational points on a curve over a finite field and provide the necessary background on ring and field theory. Four different articles are analyzed, the first of these articles gives a complete set of table showing the numbers of rational points for curves with genus up to 50. The other articles provide interesting constructions of covering curves: covers by the Hemitian curve, Kummer extensions and Artin-Schreier extensions. With these articles the great difficulty of finding explicit equations for curves with many rational points is overcome. With the method given by Arnaldo García in [6] we have been able to find examples that can be used to define the lower bounds for the corresponding entries in the tables given in http: //˜geer, which to the time of writing this Thesis appear as ”no information available”. In fact, as the curves found are maximal, these entries no longer need a bound, they can be given by a unique entry, since the exact value of Nq(g) is now known. At the end of the thesis an outline of the construction of Goppa codes is given and the NXL and XNL codes are presented.
Materia(s) Matemáticas
Editor(es) Universidad Nacional de Educación a Distancia (España). Facultad de Ciencias
Linköpings Universitet. MAI Mathematics
Director/Tutor Izquierdo, Milagros
Fecha 2010-06-01
Formato application/pdf
Identificador bibliuned:mastermatavanz20
Idioma eng
Versión de la publicación acceptedVersion
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Tipo de recurso master Thesis
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Creado: Thu, 10 Mar 2011, 13:11:45 CET