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. / 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.

Orientador: Reginaldo Palazzo Junior / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação / Made available in DSpace on 2018-08-09T06:02:14Z (GMT). No. of bitstreams: 1 Bastos_JeffersonLuizRocha_D.pdf: 792249 bytes, checksum: 7a08577655f1a651a653df4d98e29e62 (MD5) Previous issue date: 2007 / Resumo: Tendo como base trabalhos recentes que associam o desempenho de sistemas de comunicação digital ao gênero de uma superfície compacta de Riemann, este trabalho tem como objetivo propor uma integração entre modulação e codificação de canal, tendo como base o gênero da superfície. Para atingir tais objetivos, nossa proposta é a seguinte: fixado um gênero g (g = 0,1,2,3), encontrar curvas com este gênero e fazer uma análise dos parâmetros dos códigos associados a esta curva, a fim de se obter uma modulação e um sub-código desta modulação para ser utilizado na codificação de canal / Abstract: Based on recent research showing that the performance of bandwidth efficent communication systems also depends on the genus of a. compact Riemann surface in which the communication channel is embedded, this study aims at proposing a combined form of modulation and coding technique when only the genus of a surface is given to the communication system designeI. To achieve this goal, the following procedure is proposed. Knowing that the channel is embedded in a surface of genus g, find algebraic curves with the given genus which will give rise to the modulation system, an (n, n, 1) type of code, and from this find the best (n, k, d) subcode, to be employed in the aforementioned combined formo Keywords: Riemann surface, algebraic curves, Goppa codes, modulation / Doutorado / Engenharia de Computação / Doutor em Engenharia Elétrica

Orientador: Fernando Eduardo Torres Orihuela / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-11T03:53:44Z (GMT). No. of bitstreams: 1 Tizziotti_GuilhermeChaud_D.pdf: 891044 bytes, checksum: 4e90b377185548b051c39ead60bdf183 (MD5) Previous issue date: 2008 / Resumo: Estendemos resultados de Heegard, Little e Saints relacionados a bases de Gröbner para códigos Hermitianos pontuais. Trabalhamos com códigos Hermitianos bipontuais e n-pontuais, e com códigos sobre a curva Norma-Traço. Além disso, determinamos o semigrupo de Weierstrass de um certo par de pontos racionais sobre a curva Norma-Traço e com esse semigrupo conseguimos melhorar a cota da distância mínima de códigos construídos sobre tais curvas / Abstract: We extend results of Heegard, Little and Saints concerning the Gröbner basis algorithm for one-point Hermitian codes. We work with two-point and n-point Hermitian codes and codes arising from the Norm-Trace curve. We also determine the Weierstrass semigroup at a certain pair of rational points in such curves and uses these computations to improve the lower bound on the minimum distance of two-point algebraic geometry codes arising from them / Doutorado / Algebra, Geometria Algebrica / Doutor em Matemática

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. / 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.

Thesis (MSc (Mathematics))--University of Stellenbosch, 2006. / The main topic of this thesis is the construction of the algebraic geometric codes (Goppa codes), and their decoding by the list-decoding, which allows one to correct beyond half of the minimum distance. We also consider the list-decoding of the Reed–Solomon codes as they are subclass of the Goppa codes, and the determination of the parameters of the non primitive BCH codes. AMS Subject Classification: 4B05, 94B15, 94B35, 94B27, 11T71, 94B65,B70. Keywords: Linear codes, cyclic codes, BCH codes, Reed–Solomon codes, list-decoding, Algebraic Geometric codes, decoding, bound on codes, error probability.

Evaluation codes (also called order domain codes) are traditionally introduced as generalized one point geometric Goppa codes. In the present dissertation we will give a new point of view on evaluation codes by introducing them instead as particular nice examples of affine variety codes. Our study includes a reformulation of the usual methods to estimate the minimum distances of evaluation codes into the setting of affine variety codes. Finally we describe the connection to the theory of one point geometric Goppa codes. / Códigos de avaliação (também chamados códigos de domínio de ordem) são tradicionalmente apresentados como códigos de Goppa de um ponto generalizados. Na presente dissertação, vamos estudar um novo ponto de vista sobre códigos de avaliação, introduzindo-os como bons exemplos particulares de códigos de variedades afins. Nosso estudo inclui uma reformulação dos métodos usuais para estimar as distâncias mínimas de códigos de avaliação no conjunto dos códigos de variedades afins. Finalmente descrevemos a conexão com a teoria dos códigos geométricos Goppa de um ponto. / Mestre em Matemática

Orientador: Fernando Eduardo Torres Orihuela / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-10T07:01:07Z (GMT). No. of bitstreams: 1 SepulvedaCastellanos_Alonso_D.pdf: 783003 bytes, checksum: 2af4bba938cd5b7d31fcd02a5c79ac85 (MD5) Previous issue date: 2008 / Resumo: Estudamos os códigos de Goppa (códigos GH) sobre certos corpos de funções algébricas com muitos lugares racionais. Estes códigos generalizam os bem conhecidos códigos Hermitianos; portanto podemos esperar que estes códigos tenham bons parâmetros. Bulygin (IEEE Trans. Inform. Theory 52 (10), 4664¿4669 (2006)) inicia o estudo dos códigos GH; enquanto Bulygin considerou somente característica par, nosso trabalho 'e feito em qualquer característica. Em qualquer caso, nosso trabalho é fortemente influenciado pelo de Bulygin. A seguir, listamos alguns dos nossos resultados com respeito aos códigos GH. ¿ Calculamos ¿distâncias mínimas exatas¿, em particular, melhoramos os resultados de Bulygin; ¿ Encontramos cotas para os pesos generalizados de Hamming, al'em disso, mostramos um algoritmo para aplicar estes cálculos na criptografia; ¿ Calculamos um subgrupo de Automorfismos; ¿ Consideramos códigos em determinados subcorpos dos corpos usados para construir os códigos GH / Abstract: We study Goppa codes (GH codes) based on certain algebraic function fields whose number of rational places is large. These codes generalize the well-known Hermitian codes; thus we might expect that they have good parameters. Bulygin (IEEE Trans. Inform. Theory 52 (10), 4664¿4669 (2006)) initiate the study of GH-codes; while he considered only the even characteristic, our work is done regardless the characteristic. In any case our work was strongly influenced by Bulygin¿s. Next we list some of the results of our work with respect to GH-codes. ¿ We calculate ¿true minimum distances¿, in particular, we improve Bulygin¿s results; ¿ We find bounds on the generalized Hamming weights, moreover, we show an algorithm to apply these computations to the cryptography; ¿ We calculate an Automorphism subgroup; ¿ We consider codes on certain subfields of the fields used for to construct GH-codes / Doutorado / Algebra (Geometria Algebrica) / Doutor em Matemática

Title: Goppa codes and their applications Author: Bc. Jaroslav Kotil Department: Department of algebra Supervisor: prof. RNDr. Aleš Drápal, CSc., DSc. Abstract: In this diploma paper we introduce Goppa codes, describe their para- metres and inclusion in Alternant codes, which are residual Generalized Reed- Solomon codes, and Algebraic-geometry codes. Aftewards we demonstrate deco- ding of Goppa codes and introduce Wild Goppa codes. We also describe post- quantum cryptography member: McEliece cryptosystem for which no effective attacks with quantum computers are known. We outline a usage of this crypto- system with Goppa codes and describe the security of the cryptosystem together with possible attacks of which the most effective ones are based on information- set decoding. Keywords: Goppa codes, Generalized Reed-Solomon codes, Algebraic-geometry codes, Post-quantum cryptography, McEliece cryptosystem 1

<p>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 nite eld 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 nd examples that can be used to define the lower bounds for the corresponding entries in the tables given in http: //wins.uva.nl/~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 N_q(g) is now known.</p><p>At the end of the thesis an outline of the construction of Goppa codes is given and the NXL and XNL codes are presented.</p><p> </p>

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 nite eld 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 nd examples that can be used to define the lower bounds for the corresponding entries in the tables given in http: //wins.uva.nl/~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.