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Doctoral Thesis
DOI
https://doi.org/10.11606/T.55.2018.tde-18102018-084025
Document
Author
Full name
Grégory Duran Cunha
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2018
Supervisor
Committee
Borges Filho, Herivelto Martins (President)
Arakelian, Nazar
Carvalho, Cícero Fernandes de
Mirzaii, Behrooz
 
Title in English
On Weierstrass points and some properties of curves of Hurwitz type
Keywords in English
Algebraic curves
Finite fields
Goppa codes
Weierstrass semigroup
Abstract in English
This work presents several results on curves of Hurwitz type, defined over a finite field. In 1961, Tallini investigated plane irreducible curves of minimum degree containing all points of the projective plane PG(2,q) over a finite field of order q. We prove that such curves are Fq3(q2+q+1)-projectively equivalent to the Hurwitz curve of degree q+2, and compute some of itsWeierstrass points. In addition, we prove that when q is prime the curve is ordinary, that is, the p-rank equals the genus of the curve. We also compute the automorphism group of such curve and show that some of the quotient curves, arising from some special cyclic automorphism groups, are still curves of Hurwitz type. Furthermore, we solve the problem of explicitly describing the set of all Weierstrass pure gaps supported by two or three special points on Hurwitz curves. Finally, we use the latter characterization to construct Goppa codes with good parameters, some of which are current records in the Mint table.
 
Title in Portuguese
Pontos de Weierstrass e algumas propriedades das curvas do tipo Hurwitz
Keywords in Portuguese
Códigos de Goppa
Corpos finitos
Curvas algébricas
Semigrupo de Weierstrass
Abstract in Portuguese
Este trabalho apresenta vários resultados em curvas do tipo Hurwitz, definidas sobre um corpo finito. Em 1961, Tallini investigou curvas planas irredutíveis de grau mínimo contendo todos os pontos do plano projetivo PG(2,q) sobre um corpo finito de ordem q. Provamos que tais curvas são Fq3(q2+q+1)-projetivamente equivalentes à curva de Hurwitz de grau q+2, e calculamos alguns de seus pontos de Weierstrass. Em adição, provamos que, quando q é primo, a curva é ordinária, isto é, o p-rank é igual ao gênero da curva. Também calculamos o grupo de automorfismos desta curva e mostramos que algumas das curvas quocientes, construídas a partir de certos grupos cíclicos de automorfismos, são ainda curvas do tipo Hurwitz. Além disso, solucionamos o problema de descrever explicitamente o conjunto de todos os gaps puros de Weierstrass suportados por dois ou três pontos especiais em curvas de Hurwitz. Finalmente, usamos tal caracterização para construir códigos de Goppa com bons parâmetros, sendo alguns deles recordes na tabela Mint.
 
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Publishing Date
2018-10-18
 
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