On Weierstrass points and some properties of curves of Hurwitz type

• Ano de defesa: 2018
• Autor(a) principal: Cunha, Grégory Duran
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Biblioteca Digitais de Teses e Dissertações da USP
• Palavras-chave em Português: Algebraic curves; Códigos de Goppa; Corpos finitos; Curvas algébricas; Finite fields; Goppa codes; Semigrupo de Weierstrass; Weierstrass semigroup
• Link de acesso: http://www.teses.usp.br/teses/disponiveis/55/55135/tde-18102018-084025/

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.