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Dissertations / Theses on the topic 'Algebraic geometric codes'
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Guenda, Kenza. "On algebraic geometric codes and some related codes." Thesis, Stellenbosch : University of Stellenbosch, 2006. http://hdl.handle.net/10019.1/2033.
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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.
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CRISSAFF, LHAYLLA DOS SANTOS. "AN ALGEBRAIC CONSTRUCTION OF GEOMETRIC CODES." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2005. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=7082@1.
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COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIORComeçamos estudando uma classe particular de códigos lineares, os chamados códigos de Goppa que são obtidos calculando o valor de certas funções em pontos de Kn, onde K é um corpo finito. Apresentamos uma generalização desta construção e definimos códigos de avaliação sobre K- ágebras satisfazendo certas propriedades. Para estes códigos, descrevemos um algoritmo de decodificação e mostramos que se considerarmos os códigos de Goppa em um ponto como exemplo desta nova construção, o algoritmo corrige mais erros do que o algoritmo clássico para os códigos de Goppa.
We begin studying a certain type of linear code the so-called Goppa codes. These codes are constructed by taking the evaluation of certain functions at points in Kn, where K is a finite field. As a generalization of this construction, we introduce the so-called evaluation codes defined over K-algebras satisfying some properties. For these codes, we describe a decoding algorithm and we show that if we consider classical one-point Goppa codes as an example of the new construction, this algorithm correct more errors that the classical algorithm for Goppa codes.
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Iannone, Paola. "Automorphism groups of geometric codes." Thesis, University of East Anglia, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.318091.
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Johnston, Martin. "Design and implementation of algebraic-geometric codes over AWGN and fading channels." Thesis, University of Newcastle Upon Tyne, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.430341.
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Marhenke, Jörg. "On algorithms for coding and decoding algebraic-geometric codes and their implementation." [S.l. : s.n.], 2008. http://nbn-resolving.de/urn:nbn:de:bsz:289-vts-65822.
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Alkandari, Maryam Mohammed. "Decoding partial algebraic geometrric codes." Thesis, Imperial College London, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.405512.
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Jeffs, Robert Amzi. "Convexity of Neural Codes." Scholarship @ Claremont, 2016. https://scholarship.claremont.edu/hmc_theses/87.
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An important task in neuroscience is stimulus reconstruction: given activity in the brain, what stimulus could have caused it? We build on previous literature which uses neural codes to approach this problem mathematically. A neural code is a collection of binary vectors that record concurrent firing of neurons in the brain. We consider neural codes arising from place cells, which are neurons that track an animal's position in space. We examine algebraic objects associated to neural codes, and completely characterize a certain class of maps between these objects. Furthermore, we show that such maps have natural geometric implications related to the receptive fields of place cells. Lastly we describe several purely geometric results related to neural codes.
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Rocha, Junior Mauro Rodrigues. "Bases de Gröbner aplicadas a códigos corretores de erros." Universidade Federal de Juiz de Fora (UFJF), 2017. https://repositorio.ufjf.br/jspui/handle/ufjf/5946.
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O principal objetivo desse trabalho é estudar duas aplicações distintas das bases de Gröbner a códigos lineares. Com esse objetivo, estudamos como relacionar códigos a outras estruturas matemáticas, fazendo com que tenhamos novas ferramentas para a realização da codificação. Em especial, estudamos códigos cartesianos afins e os códigos algébrico-geométricos de Goppa.
The main objective of this work is to study two different applications of Gröbner basis to linear codes. With this purpose, we study how to relate codes to other mathematical structures, allowing us to use new tools to do the coding. In particular, we study affine cartesian codes e algebraic-geometric Goppa codes.
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Berardini, Elena. "Algebraic geometry codes from surfaces over finite fields." Thesis, Aix-Marseille, 2020. http://www.theses.fr/2020AIXM0170.
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Nous proposons, dans cette thèse, une étude théorique des codes géométriques algébriques construits à partir de surfaces définies sur les corps finis. Nous prouvons des bornes inférieures pour la distance minimale des codes sur des surfaces dont le diviseur canonique est soit nef soit anti-strictement nef et sur des surfaces sans courbes irréductibles de petit genre. Nous améliorons ces bornes inférieures dans le cas des surfaces dont le nombre de Picard arithmétique est égal à un, des surfaces sans courbes de petite auto-intersection et des surfaces fibrées. Ensuite, nous appliquons ces bornes aux surfaces plongées dans P3. Une attention particulière est accordée aux codes construits à partir des surfaces abéliennes. Dans ce contexte, nous donnons une borne générale sur la distance minimale et nous démontrons que cette estimation peut être améliorée en supposant que la surface abélienne ne contient pas de courbes absolument irréductibles de petit genre. Dans cette optique nous caractérisons toutes les surfaces abéliennes qui ne contiennent pas de courbes absolument irréductibles de genre inférieur ou égal à 2. Cette approche nous conduit naturellement à considérer les restrictions de Weil de courbes elliptiques et les surfaces abéliennes qui n'admettent pas de polarisation principaleIn this thesis we provide a theoretical study of algebraic geometry codes from surfaces defined over finite fields. We prove lower bounds for the minimum distance of codes over surfaces whose canonical divisor is either nef or anti-strictly nef and over surfaces without irreducible curves of small genus. We sharpen these lower bounds for surfaces whose arithmetic Picard number equals one, surfaces without curves with small self-intersection and fibered surfaces. Then we apply these bounds to surfaces embedded in P3. A special attention is given to codes constructed from abelian surfaces. In this context we give a general bound on the minimum distance and we prove that this estimation can be sharpened under the assumption that the abelian surface does not contain absolutely irreducible curves of small genus. In this perspective we characterize all abelian surfaces which do not contain absolutely irreducible curves of genus up to 2. This approach naturally leads us to consider Weil restrictions of elliptic curves and abelian surfaces which do not admit a principal polarization
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Rovi, Carmen. "Algebraic Curves over Finite Fields." Thesis, Linköping University, Department of Mathematics, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-56761.
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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.
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Bastos, Jefferson Luiz Rocha. "Forma combinada de conjunto de sinais e codigos de Goppa atraves da geometria algebrica." [s.n.], 2007. http://repositorio.unicamp.br/jspui/handle/REPOSIP/261299.
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Orientador: Reginaldo Palazzo JuniorTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação
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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
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Chaussade, Lionel. "Codes correcteurs avec les polynômes tordus." Phd thesis, Université Rennes 1, 2010. http://tel.archives-ouvertes.fr/tel-00813705.
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Les anneaux de polynômes sont l'un des outils privilégiés pour construire et étudier des familles de codes correcteurs. Nous nous proposons, dans cette thèse, d'utiliser des anneaux de Öre, qui sont des anneaux de polynômes non-commutatifs, afin de créer des codes correcteurs. Cette approche nous permet d'obtenir des familles de codes correcteurs plus larges que si l'on se restreint au cas commutatif mais qui conservent de nombreuses propriétés communes. Nous obtenons notamment un algorithme qui permet de fabriquer des codes correcteurs dont la distance de Hamming ou la distance rang est prescrite. C'est ainsi que nous avons exhibé deux codes qui améliorent la meilleure distance minimale connue pour un code de même longueur et de même dimension. L'un est de paramètres $[42,14,21]$ sur le corps $\mathbb{F}_8$ et l'autre de paramètres $[40,23,10]$ sur $\mathbb{F}_4$. La généralisation de cette étude au cas d'anneaux polynomiaux multivariés est également présentée; l'outil principal est alors la théorie des bases de Gröbner qui s'adapte dans ce cadre non-commutatif et permet de manipuler des idéaux pour créer de nouvelles familles de codes correcteurs.
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Silva, Pryscilla dos Santos Ferreira. "Códigos lineares disjuntos e corpos de funções algébricas." Universidade Federal da Paraíba, 2011. http://tede.biblioteca.ufpb.br:8080/handle/tede/7350.
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Previous issue date: 2011-02-24Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
In this work, based on algebraic function fields, we give constructions of disjoint linear codes. In addition,we study the asymptotic behavior of disjoint linear codes from our constructions.
Neste trabalho, baseados em corpos de funções algébricas, forneceremos construções de códigos lineares disjuntos. Além disso, nós estudaremos comportamentos assintóticos de códigos lineares disjuntos a partir da nossa construção.
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Melo, Nolmar. "Codigos geometricos de Goppa via metodos elementares." [s.n.], 2006. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306316.
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Orientadores: Paulo Roberto Brumatti, Fernando Eduardo Torres OrihuelaDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: O objetivo central desta dissertação foi o de apresentar os Códigos Geométricos de Goppa via métodos elementares que foram introduzidos por J. H. van Lint, R. Pellikaan e T. Hfhold por volta de 1998. Numa primeira parte da dissertação são apresentados os conceitos fundamentais sobre corpos de funções racionais de uma curva algébrica na direção de se definir os códigos de Goppa de maneira clássica, neste estudo nos baseamos principalmente no livro ¿Algebraic Function Fields and Codes¿ de H. Stichtenoth. A segunda parte inicia-se com a introdução dos conceitos de funções peso, grau e ordem que são fundamentais para o estudo dos Códigos de Goppa via métodos elementares de álgebra linear e de semigrupos, tal estudo foi baseado em ¿Algebraic geometry codes¿ de J. H. van Lint, R. Pellikaan e T. Hfhold.A dissertação termina com a apresentação de exemplos que ilustram os métodos elementares que nos referimos acima
Abstract: The central objective of this dissertation was to present the Goppa Geometry Codes via elementary methods which were introduced by J. H. van Lint, R. Pellikaan and T. Hfhold about 1998. On the first past of such dissertation are presented the fundamental concepts about fields of rational functions of an algebraic curve in the direction as to define the Goppa Codes on a classical manner. In this study we based ourselves mainly on the book ¿Algebraic Function Fields and Codes¿ of H. Stichtenoth. The second part is initiated with an introduction about the functions weight, degree and order which are fundamental for the study of the Goppa Codes throught elementary methods of linear algebra and of semigroups and such study was based on ¿Algebraic Geometry Codes¿ of J. h. van Lint, R. Pellikaan and T. Hfhold. The dissertation ends up with a presentation of examples which illustrate the elementary methods that we have referred to above
Mestrado
Algebra
Mestre em Matemática
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Nardi, Jade. "Quelques retombées de la géométrie des surfaces toriques sur un corps fini sur l'arithmétique et la théorie de l'information." Thesis, Toulouse 3, 2019. http://www.theses.fr/2019TOU30051.
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Cette thèse, à la frontière entre les mathématiques et l'informatique, est consacrée en partie à l'étude des paramètres et des propriétés des codes de Goppa sur les surfaces de Hirzebruch. D'un point de vue arithmétique, la théorie des codes correcteurs a ravivé la question du nombre de points rationnels d'une variété définie sur un corps fini, qui semblait résolue par la formule de Lefschetz. La distance minimale de codes géométriques donne un majorant du nombre de points rationnels d'une hypersurface d'une variété donnée et de classe de Picard fixée. Ce majorant étant le plus souvent atteint pour les courbes très réductibles, il est naturel de se concentrer sur les courbes irréductibles pour affiner les bornes. On présente une stratégie globale pour majorer le nombre de points d'une variété en fonction de son ambiant et d'invariants géométriques, notamment liés à la théorie de l'intersection. De plus, une méthode de ce type pour les courbes d'une surface torique est développée en adaptant l'idée de F.J Voloch et K.O. Sthör aux variétés toriques. Enfin, on s'intéresse aux protocoles de Private Information Retrivial, qui visent à assurer qu'un utilisateur puisse accéder à une entrée d'une base de données sans révéler d'information sur l'entrée au propriétaire de la base de données. Un protocole basé sur des codes sur des plans projectifs pondérés est proposé ici. Il améliore les protocoles existants en résistant à la collusion de serveurs, au prix d'une grande perte de capacité de stockage. On pallie ce problème grâce à la méthode du lift qui permet la construction de familles de codes asymptotiquement bonnes, avec les mêmes propriétés localesA part of this thesis, at the interface between Computer Science and Mathematics, is dedicated to the study of the parameters ans properties of Goppa codes over Hirzebruch surfaces. From an arithmetical perspective, the question about number of rational points of a variety defined over a finite field, which seemed dealt with by Lefchetz formula, regained interest thanks to error correcting codes. The minimum distance of an algebraic-geometric codes provides an upper bound of the number of rational points of a hypersurface of a given variety and with a fixed Picard class. Since reducible curves are most likely to reach this bound, one can focus on irreducible curves to get sharper bounds. A global strategy to bound the number of points on a variety depending on its ambient space and some of its geometric invariants is exhibited here. Moreover we develop a method for curves on toric surfaces by adapting F.J. Voloch et K.O. Sthör's idea on toric varieties. Finally, we interest in Private Information Retrivial protocols, which aim to ensure that a user can access an entry of a database without revealing any information on it to the database owner. A PIR protocol based on codes over weighted projective planes is displayed here. It enhances other protocols by offering a resistance to servers collusions, at the expense of a loss of storage capacity. This issue is fixed by a lifting process, which leads to asymptotically good families of codes, with the same local properties
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Koshelev, Dmitrii. "Nouvelles applications des surfaces rationnelles et surfaces de Kummer généralisées sur des corps finis à la cryptographie à base de couplages et à la théorie des codes BCH." Thesis, université Paris-Saclay, 2021. http://www.theses.fr/2021UPASM001.
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Il y a une théorie bien développée de ce qu'on appelle codes toriques, c'est-à-dire des codes de géométrie algébrique sur des variétés toriques sur un corps fini. A côté des tores et variétés toriques ordinaires (c'est-à-dire déployés), il y a non-déployés. La thèse est donc dédiée à l'étude des codes de géométrie algébrique sur les derniersThere is well developed theory of so-called toric codes, i.e., algebraic geometry codes on toric varieties over a finite field. Besides ordinary (i.e., split) tori and toric varieties there are non-split ones. Therefore the thesis is dedicated to the study of algebraic geometry codes on the latter
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Caullery, Florian. "Polynomes sur les corps finis pour la cryptographie." Thesis, Aix-Marseille, 2014. http://www.theses.fr/2014AIXM4013/document.
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Les fonctions de F_q dans lui-même sont des objets étudiés dans de divers domaines tels que la cryptographie, la théorie des codes correcteurs d'erreurs, la géométrie finie ainsi que la géométrie algébrique. Il est bien connu que ces fonctions sont en correspondance exacte avec les polynômes en une variable à coefficients dans F_q. Nous étudierons trois classes de polynômes particulières: les polynômes Presque Parfaitement Non linéaires (Almost Perfect Nonlinear (APN)), les polynômes planaires ou parfaitement non linéaire (PN) et les o-polynômes.Les fonctions APN sont principalement étudiées pour leurs applications en cryptographie. En effet, ces fonctions sont celles qui offre la meilleure résistance contre la cryptanalyse différentielle.Les polynômes PN et les o-polynômes sont eux liés à des problèmes célèbres de géométrie finie. Les premiers décrivent des plans projectifs et les seconds sont en correspondance directe avec les ovales et hyperovales de P^2(F_q). Néanmoins, leurs champ d'application a été récemment étendu à la cryptographie symétrique et à la théorie des codes correcteurs d'erreurs.L'un des moyens utilisé pour compléter la classification est de considérer les polynômes présentant l'une des propriétés recherchées sur une infinité d'extension de F_q. Ces fonctions sont appelées fonction APN (respectivement PN ou o-polynômes) exceptionnelles.Nous étendrons la classification des polynômes APN et PN exceptionnels et nous donneront une description complète des o-polynômes exceptionnels. Les techniques employées sont basées principalement sur la borne de Lang-Weil et sur des méthodes élémentairesFunctions from F_q to itself are interesting objects arising in various domains such as cryptography, coding theory, finite geometry or algebraic geometry. It is well known that these functions admit a univariate polynomial representation. There exists many interesting classes of such polynomials with plenty of applications in pure or applied maths. We are interested in three of them: Almost Perfect Nonlinear (APN) polynomials, Planar (PN) polynomials and o-polynomials. APN polynomials are mostly used in cryptography to provide S-boxes with the best resistance to differential cryptanalysis and in coding theory to construct double error-correcting codes. PN polynomials and o-polynomials first appeared in finite geometry. They give rise respectively to projective planes and ovals in P^2(F_q). Also, their field of applications was recently extended to symmetric cryptography and error-correcting codes.A complete classification of APN, PN and o-polynomials is an interesting open problem that has been widely studied by many authors. A first approach toward the classification was to consider only power functions and the studies were recently extended to polynomial functions.One way to face the problem of the classification is to consider the polynomials that are APN, PN or o-polynomials over infinitely many extensions of F_q, namely, the exceptional APN, PN or o-polynomials.We improve the partial classification of exceptional APN and PN polynomials and give a full classification of exceptional o-polynomials. The proof technique is based on the Lang-Weil bound for the number of rational points in algebraic varieties together with elementary methods
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"The development of algebraic-geometric codes & their applications." 1999. http://library.cuhk.edu.hk/record=b5890065.
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by Ho Kin Ming.Thesis (M.Phil.)--Chinese University of Hong Kong, 1999.
Includes bibliographical references (leaves 68-69).
Abstracts in English and Chinese.
Chapter 0 --- Introduction --- p.5
Chapter 1 --- Introduction to Coding Theory --- p.9
Chapter 1.1 --- Definition of a code --- p.10
Chapter 1.2 --- Maximum Likelihood Decoding --- p.11
Chapter 1.3 --- Syndrome Decoding --- p.12
Chapter 1.4 --- Two Kinds of Errors and Concatenated Code --- p.14
Chapter 2 --- Basic Knowledge of Algebraic Curve --- p.16
Chapter 2.1 --- Affine and Projective Curve --- p.16
Chapter 2.2 --- Regular Functions and Maps --- p.17
Chapter 2.3 --- Divisors and Differential forms --- p.19
Chapter 2.4 --- Riemann-Roch Theorem --- p.21
Chapter 3 --- Construction of Algebraic Geometric Code --- p.23
Chapter 3.1 --- L-construction --- p.23
Chapter 3.2 --- Ω -construction --- p.24
Chapter 3.3 --- Duality --- p.26
Chapter 4 --- Basic Error Processing --- p.28
Chapter 4.1 --- Error Locators and Syndromes --- p.28
Chapter 4.2 --- Finding an Error Locator --- p.29
Chapter 5 --- Full Error Processing for Code on Curve of Genus1 --- p.34
Chapter 5.1 --- Syndrome table --- p.34
Chapter 5.2 --- Syndrome table --- p.36
Chapter 5.3 --- The algorithm of Full Error Processing --- p.38
Chapter 5.4 --- A simple Example --- p.40
Chapter 6 --- General Full Error Processing --- p.47
Chapter 6.1 --- Row Candidate and Column Candidate --- p.47
Chapter 6.2 --- Consistency --- p.49
Chapter 6.3 --- Majority voting --- p.50
Chapter 6.4 --- Example --- p.53
Chapter 7 --- Application of Algebraic Geometric Code --- p.60
Chapter 7.1 --- Communication --- p.60
Chapter 7.2 --- Cryptosystem --- p.62
Bibliography
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Liu, Chih-Wei, and 劉志尉. "A Simple and Efficient Decoding Algorithm for Algebraic-Geometric Codes." Thesis, 1999. http://ndltd.ncl.edu.tw/handle/69141351180632628012.
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博士國立清華大學
電機工程學系
87
By adopting the extended syndrome matrix M with a restricted Gaussian elimination, a simple and efficient decoding algorithm for algebraic-geometric codes is developed. The decoding algorithm can be considered as the refinement of the Feng-Rao algorithm and can be implemented by a set of r parallel early stopped Berlekamp-Massey algorithms, where r is the smallest nonzero nongap of the algebraic-geometric curve over which the code is defined. The computation complexity of the algorithm is in the order of O(r n^2), which is the same as that of the Kotter''s algorithm, where n is the code length. Comparing with the Kotter''s algorithm, the proposed decoding algorithm is superior in the following aspects. Firstly, with the early stopped property, the proposed algorithm can save both processing time and computation complexity. In particular, for decoding (n, n-2t) BCH codes, i.e. r=1, the proposed algorithm requires only t+e iterations (or steps) to determine the error-locator polynomial, where e is the number of errors actually occurred. While, the Kotter''s algorithm requires the constant 2t iterations. Secondly, with storing both nonzero discrepancy as well as the corresponding coefficient vector, the proposed algorithm prevents from the additional multiplicative operations for the normalization of the saved coefficient vector. The saved coefficient vector needs to be normalized only when it is being used to update the currently used coefficient vector. And finally, an accurate method of counting the available candidates is developed in the algorithm. Based on the point of view from the Feng-Rao algorithm, the method to count the total number of the available candidates is not correct in that of the Kotter''s algorithm.
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Gu, Wei-Hsin, and 辜維欣. "A Construction of Good Algebraic Geometric Codes Based on Some Towers of Algebraic Function Fields." Thesis, 2001. http://ndltd.ncl.edu.tw/handle/99951865264133118879.
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碩士國立清華大學
電機工程學系
89
In this thesis, we investigate a special tower of function fields. The primary goal is to construct a class of algebraic geometric codes based on this tower. We will demonstrate that, although this tower is asymptotically bad, the algebraic geometric codes corresponding to the ith function field of this tower are better than Reed-Solomon codes and Hermitian codes in many practical channels when i is small. Finally, we give an easy construction of these codes.
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Marhenke, Jörg [Verfasser]. "On algorithms for coding and decoding algebraic-geometric codes and their implementation / vorgelegt von Jörg Marhenke." 2008. http://d-nb.info/998489654/34.
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Huang, Kuo Tai, and 黃國泰. "A systolic array architecture for the decoding of algebraic- geometric codes with modified Feng-Rao algorithm." Thesis, 1996. http://ndltd.ncl.edu.tw/handle/95458916818186210050.
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碩士國立清華大學
電機工程研究所
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Feng-Rao algorithm is a successful algorithm for the decoding of algebraic-geometric (AG) codes. However, there is no implementation of this algorithm up to now. In this thesis, we have modified the Feng-Rao algorithm to have more parallelism and developed a systolic array architecture for VLSI implementation. The symmetry property of the syndrome matrix has been exploited to reduce the complexity of this architecture. The complexity of our proposed systolic array architecture is t^3/6+(1+g')t^2/2+[(g-3)g'/2-2/3+g]t, which is comparable to that elimination on a square matrix with matrix size equal to t, where t is the error-correcting capability of a code, g is the genus of the curve, and g'=\floor(g-1/2). The control circuit in oursimple. Besides, we have also proposed a circuitry to perform the majority voting scheme needed in the Feng-Rao algorithm with the consideration that the candidates are q-ary symbols.
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Tsao, Shih-chang. "On explicit constructions and improved bounds of algebraic geometric code." 2006. http://etda.libraries.psu.edu/theses/approved/WorldWideIndex/ETD-1366/index.html.
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Heglasová, Veronika. "Algebraicko-geometrické kódy a Gröbnerovy báze." Master's thesis, 2013. http://www.nusl.cz/ntk/nusl-324636.
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In this master thesis we introduce algebraic geometry codes (AG codes). Be- sides basic definitions, properties and attributes of AG codes and algebraic ge- ometry we show how to encode AG codes that has nontrivial Abelian group of permutation automorphisms and how to decode one-point AG codes. We also present Hermitian codes, which are example of one-point AG codes with nontriv- ial Abelian group of permutation automorphisms. We demonstrate the method for encoding and the method for decoding on specific Hermitian code. 1
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"Caracterizações algebrica e geometrica dos codigos propelineares." Tese, Biblioteca Digital da Unicamp, 2000. http://libdigi.unicamp.br/document/?code=vtls000206240.
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Rytíř, Pavel. "Geometrické a algebraické vlastnosti diskrétních struktur." Doctoral thesis, 2013. http://www.nusl.cz/ntk/nusl-322617.
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In the thesis we study two dimensional simplicial complexes and linear codes. We say that a linear code C over a field F is triangular representable if there exists a two dimensional simplicial complex ∆ such that C is a punctured code of the kernel ker ∆ of the incidence matrix of ∆ over F and dim C = dim ker ∆. We call this simplicial complex a geometric representation of C. We show that every linear code C over a primefield is triangular representable. In the case of finite primefields we construct a geometric representation such that the weight enumerator of C is obtained by a simple formula from the weight enumerator of the cycle space of ∆. Thus the geometric representation of C carries its weight enumerator. Our motivation comes from the theory of Pfaffian orientations of graphs which provides a polynomial algorithm for weight enumerator of the cut space of a graph of bounded genus. This algorithm uses geometric properties of an embedding of the graph into an orientable Riemann surface. Viewing the cut space of a graph as a linear code, the graph is thus a useful geometric representation of this linear code. We study embeddability of the geometric representations into Euclidean spaces. We show that every binary linear code has a geometric representation that can be embed- ded into R4 . We characterize...
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Kotil, Jaroslav. "Goppovy kódy a jejich aplikace." Master's thesis, 2013. http://www.nusl.cz/ntk/nusl-321400.
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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
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"Caracterizações topologica, geometrica e algebrica dos produtos da recombinação do DNA atraves dos modelos Tangle e frações continuas." Tese, Biblioteca Digital da Unicamp, 2004. http://libdigi.unicamp.br/document/?code=vtls000335214.
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Ethamakula, Bharath Kumar. "Asymptotic Lower Bound for Quasi Transitive Codes over Cubic Finite Fields." Thesis, 2015. http://etd.iisc.ernet.in/2005/3821.
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Algebraic geometric codes were first introduced by V.D.Goppa . They were well recognized and developed by Tsfasman, Vladut and Zink because they have parameters better than Gilbert-Varshmov bound and thus giving rise to Tsfasman Vladut-Zink bound. While the codes given by Ihara, Tsfasman, Vladut and Zink have complicated construction, Garcia and Stichtenoth on the other hand gave an explicit construction of codes attaining Tsfasman-Vlasut-Zink bound using the terminology of function fields. In coding theory one of the challenging problem is to find a sequence of cyclic codes that are asymptotically good. While this has not been achieved, Stichtenoth generalized cyclic codes to transitive codes and constructed a sequence of asymptotically good transitive codes on algebraic function fields over quadratic finite fields that attain Tsfasman-Vladut-Zink bound.
In the case of cubic finite fields, Bezerra, Garcia and Stichtenoth constructed a tower of function fields over cubic finite fields whose limit attains a lower bound and the codes constructed over this tower turns out to be asymptotically good attaining a positive lower bound. Bassa used this tower and constructed quasi transitive codes which are a generalization of transitive codes and proved that they are also asymptotically good and attain the same positive lower bound. Later Bassa, Garcia and Stichtenoth constructed a new tower of function fields over cubic finite fields whose structure is less complicated compared to that of Bezerra, Garcia and Stichtenoths' and proved that codes constructed over it also attain the same positive lower bound. In this work along the lines of Bassa and Stichtenoth we construct quasi transitive codes over the tower given by Bassa, Garcia and Stichtenoth and prove that these quasi transitive codes are also asymptotically good and also attain the same lower bound.
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Marchal, Olivier. "Aspects géométriques et intégrables des modèles de matrices aléatoires." Thèse, 2010. http://hdl.handle.net/1866/6861.
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Cette thèse traite des aspects géométriques et d'intégrabilité associés aux modèles de matrices aléatoires. Son but est de présenter diverses applications des modèles de matrices aléatoires allant de la géométrie algébrique aux équations aux dérivées partielles des systèmes intégrables. Ces différentes applications permettent en particulier de montrer en quoi les modèles de matrices possèdent une grande richesse d'un point de vue mathématique.
Ainsi, cette thèse abordera d'abord l'étude de la jonction de deux intervalles du support de la densité des valeurs propres au voisinage d'un point singulier. On montrera plus précisément en quoi ce régime limite particulier aboutit aux équations universelles de la hiérarchie de Painlevé II des systèmes intégrables. Ensuite, l'approche des polynômes (bi)-orthogonaux, introduite par Mehta pour le calcul des fonctions de partition, permettra d'énoncer des problèmes de Riemann-Hilbert et d'isomonodromies associés aux modèles de matrices, faisant ainsi le lien avec la théorie de Jimbo-Miwa-Ueno. On montrera en particulier que le cas des modèles à deux matrices hermitiens se transpose à un cas dégénéré de la théorie isomonodromique de Jimbo-Miwa-Ueno qui sera alors généralisé. La méthode des équations de boucles avec ses notions centrales de courbe spectrale et de développement topologique permettra quant à elle de faire le lien avec les invariants symplectiques de géométrie algébrique introduits récemment par Eynard et Orantin. Ce dernier point fera également l'objet d'une généralisation aux modèles de matrices non-hermitien (beta quelconque) ouvrant ainsi la voie à la ``géométrie algébrique quantique'' et à la généralisation de ces invariants symplectiques pour des courbes ``quantiques''. Enfin, une dernière partie sera consacrée aux liens étroits entre les modèles de matrices et les problèmes de combinatoire. En particulier, l'accent sera mis sur les aspects géométriques de la théorie des cordes topologiques avec la construction explicite d'un modèle de matrices aléatoires donnant le dénombrement des invariants de Gromov-Witten pour les variétés de Calabi-Yau toriques de dimension complexe trois utilisées en théorie des cordes topologiques.
L'étendue des domaines abordés étant très vaste, l'objectif de la thèse est de présenter de façon la plus simple possible chacun des domaines mentionnés précédemment et d'analyser en quoi les modèles de matrices peuvent apporter une aide précieuse dans leur résolution. Le fil conducteur étant les modèles matriciels, chaque partie a été conçue pour être abordable pour un spécialiste des modèles de matrices ne connaissant pas forcément tous les domaines d'application présentés ici.This thesis deals with the geometric and integrable aspects associated with random matrix models. Its purpose is to provide various applications of random matrix theory, from algebraic geometry to partial differential equations of integrable systems. The variety of these applications shows why matrix models are important from a mathematical point of view. First, the thesis will focus on the study of the merging of two intervals of the eigenvalues density near a singular point. Specifically, we will show why this special limit gives universal equations from the Painlevé II hierarchy of integrable systems theory. Then, following the approach of (bi) orthogonal polynomials introduced by Mehta to compute partition functions, we will find Riemann-Hilbert and isomonodromic problems connected to matrix models, making the link with the theory of Jimbo, Miwa and Ueno. In particular, we will describe how the hermitian two-matrix models provide a degenerate case of Jimbo-Miwa-Ueno's theory that we will generalize in this context. Furthermore, the loop equations method, with its central notions of spectral curve and topological expansion, will lead to the symplectic invariants of algebraic geometry recently proposed by Eynard and Orantin. This last point will be generalized to the case of non-hermitian matrix models (arbitrary beta) paving the way to ``quantum algebraic geometry'' and to the generalization of symplectic invariants to ``quantum curves''. Finally, this set up will be applied to combinatorics in the context of topological string theory, with the explicit computation of an hermitian random matrix model enumerating the Gromov-Witten invariants of a toric Calabi-Yau threefold. Since the range of the applications encountered is large, we try to present every domain in a simple way and explain how random matrix models can bring new insights to those fields. The common element of the thesis being matrix models, each part has been written so that readers unfamiliar with the domains of application but familiar with matrix models should be able to understand it.
Travail réalisé en cotutelle avec l'université Paris-Diderot et le Commissariat à l'Energie Atomique sous la direction de John Harnad et Bertrand Eynard.