Coding Theory and Algebraic Geometry

Proceedings of the International Workshop held in Luminy, France, June 17-21, 1991

Author: Henning Stichtenoth,Michael A. Tsfasman

Publisher: Springer

ISBN: 3540472673

Category: Mathematics

Page: 232

View: 6512

About ten years ago, V.D. Goppa found a surprising connection between the theory of algebraic curves over a finite field and error-correcting codes. The aim of the meeting "Algebraic Geometry and Coding Theory" was to give a survey on the present state of research in this field and related topics. The proceedings contain research papers on several aspects of the theory, among them: Codes constructed from special curves and from higher-dimensional varieties, Decoding of algebraic geometric codes, Trace codes, Exponen- tial sums, Fast multiplication in finite fields, Asymptotic number of points on algebraic curves, Sphere packings.
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Algebra for Secure and Reliable Communication Modeling

Author: Mustapha Lahyane, Edgar Martínez-Moro

Publisher: American Mathematical Soc.

ISBN: 1470410184

Category: Geometry, Algebraic

Page: 240

View: 8096

This volume contains the proceedings of the CIMPA Research School and Conference on Algebra for Secure and Reliable Communication Modeling, held from October 1-13, 2012, in Morelia, State of Michoacán, Mexico. The papers cover several aspects of the theory of coding theory and are gathered into three categories: general theory of linear codes, algebraic geometry and coding theory, and constacyclic codes over rings. The aim of this volume is to fill the gap between the theoretical part of algebraic geometry and the applications to problem solving and computational modeling in engineering, signal processing and information theory. This book is published in cooperation with Real Sociedad Matemática Española (RSME).
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Algebraic Geometry in Coding Theory and Cryptography

Author: Harald Niederreiter,Chaoping Xing

Publisher: Princeton University Press

ISBN: 9781400831302

Category: Mathematics

Page: 272

View: 4667

This textbook equips graduate students and advanced undergraduates with the necessary theoretical tools for applying algebraic geometry to information theory, and it covers primary applications in coding theory and cryptography. Harald Niederreiter and Chaoping Xing provide the first detailed discussion of the interplay between nonsingular projective curves and algebraic function fields over finite fields. This interplay is fundamental to research in the field today, yet until now no other textbook has featured complete proofs of it. Niederreiter and Xing cover classical applications like algebraic-geometry codes and elliptic-curve cryptosystems as well as material not treated by other books, including function-field codes, digital nets, code-based public-key cryptosystems, and frameproof codes. Combining a systematic development of theory with a broad selection of real-world applications, this is the most comprehensive yet accessible introduction to the field available. Introduces graduate students and advanced undergraduates to the foundations of algebraic geometry for applications to information theory Provides the first detailed discussion of the interplay between projective curves and algebraic function fields over finite fields Includes applications to coding theory and cryptography Covers the latest advances in algebraic-geometry codes Features applications to cryptography not treated in other books
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Applied Algebra, Algebraic Algorithms and Error-Correcting Codes

18th International Symposium, AAECC-18, Tarragona, Sapin, June 8-12, 2009, Proceedings

Author: Maria Bras-Amorós,Tom Høholdt

Publisher: Springer Science & Business Media

ISBN: 3642021808

Category: Computers

Page: 243

View: 9830

The AAECC symposia serieswas started in 1983by Alain Poli (Toulouse), who, together with R. Desq, D. Lazardand P. Camion, organizedthe ?rst conference. OriginallytheacronymAAECCstoodfor"AppliedAlgebraandError-Correcting Codes."Overtheyearsitsmeaninghasshiftedto"AppliedAlgebra, Algebraic- gorithmsandError-CorrectingCodes,"re?ectingthegrowingimportanceofc- plexity, particularlyfor decoding algorithms.During the AAECC-12 symposium the ConferenceCommitteedecidedtoenforcethe theoryandpracticeofthe c- ing side as well as the cryptographic aspects. Algebra was conserved, as in the past, but slightly more oriented to algebraic geometry codes, ?nite ?elds, c- plexity, polynomials, andgraphs. The main topics for AAECC-18 were algebra, algebraiccomputation, codes and algebra, codes and combinatorics, modulation and codes, sequences, and cryptography. TheinvitedspeakersofthiseditionwereIwanDuursma, HenningStichtenoth, and Fernando Torres. We would like to express our deep regret for the loss of Professor Ralf Kotter, ] who recently passed away and could not be our fourth invited speaker. Except for AAECC-1 (Discrete Mathematics 56, 1985) and AAECC-7 (D- crete Applied Mathematics 33, 1991), the proceedings of all the symposia have been published in Springer'sLecture Notes in Computer Science (Vols. 228,229, 307, 356, 357, 508, 539, 673, 948, 1255, 1719, 2227, 2643, 3857, 4851). Itis apolicy ofAAECCto maintaina highscienti?c standard, comparableto that of a journal. This was made possible thanks to the many referees involved. Each submitted paper was evaluated by at least two international researchers. AAECC-18 received and refereed 50 submissions. Of these, 22 were selected for publication in these proceedings as regular papers and 7 were selected as extended abstracts.
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Topics in Geometry, Coding Theory and Cryptography

Author: Arnaldo Garcia,Henning Stichtenoth

Publisher: Springer Science & Business Media

ISBN: 1402053347

Category: Mathematics

Page: 201

View: 8222

The theory of algebraic function fields over finite fields has its origins in number theory. However, after Goppa`s discovery of algebraic geometry codes around 1980, many applications of function fields were found in different areas of mathematics and information theory. This book presents survey articles on some of these new developments. The topics focus on material which has not yet been presented in other books or survey articles.
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Algebraic-Geometric Codes

Author: M. Tsfasman,S.G. Vladut

Publisher: Springer Science & Business Media

ISBN: 9401138109

Category: Mathematics

Page: 667

View: 4637

'Et moi ..., si j'avait su comment en revenir, One service mathematics has rendered the je n'y serais point aIle.' human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non sense'. The series is divergent; therefore we may be able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d' etre of this series.
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Algebraic Geometry and Its Applications

Руды Ордена Ленина Математического Института Имени В.А.Стеклова

Author: Sergeĭ Mikhaĭlovich Nikolʹskiĭ,E. A. Volkov

Publisher: American Mathematical Soc.

ISBN: 9780821830925

Category: Mathematics

Page: 251

View: 2836

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Algebraic geometry and its applications

collections of papers from Shreeram S. Abhyankar's 60th birthday conference

Author: Shreeram Shankar Abhyankar,Chanderjit Bajaj

Publisher: Springer

ISBN: 9783540941767

Category: Mathematics

Page: 536

View: 1272

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Algebraic Geometry and its Applications

Proceedings of the 8th Algebraic Geometry Conference, Yaroslavl’ 1992. A Publication from the Steklov Institute of Mathematics. Adviser: Armen Sergeev

Author: Alexander Tikhomirov,Andrej Tyurin

Publisher: Springer Science & Business Media

ISBN: 3322993426

Category: Technology & Engineering

Page: 251

View: 8220

This volume contains 18 papers at the Algebraic Geometry Conference, Yaroslavl', August 10-14, 1992. These conferences in algebraic geometry have a great tradition in Russia and are helt since 1979 in Yaroslavl' every second year. The present conference, the eighth one, was the first in which several foreign mathematicians participated. From the Russian side, there was a large group of specialists in algebraic geometry and related fields (invariant theory, topology of manifolds, theory of categories, mathematical physics etc.). Lectures on modern directions in algebraic geometry, such as the theory of exceptional bundles and helices on algebraic varieties, moduli of vector bundles on algebraic surfaces with applications to Donaldson's theory, geometry of Hilbert schemes of points, twistor spaces and applications to string theory, and more traditional areas, such as birational geometry of manifolds, adjunction theory, Hodge theory, problems of rationality in the invariant theory, topology of complex algebraic varieties, and others are contained in this volume.
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Finite Fields with Applications to Coding Theory, Cryptography and Related Areas

Proceedings of the Sixth International Conference on Finite Fields and Applications, held at Oaxaca, México, May 21–25, 2001

Author: Gary L. Mullen,Henning Stichtenoth,Horacio Tapia-Recillas

Publisher: Springer Science & Business Media

ISBN: 3642594352

Category: Mathematics

Page: 335

View: 7037

The Sixth International Conference on Finite Fields and Applications, Fq6, held in the city of Oaxaca, Mexico, from May 21-25, 2001, continued a series of biennial international conferences on finite fields. This volume documents the steadily increasing interest in this topic. Finite fields are an important tool in discrete mathematics and its applications cover algebraic geometry, coding theory, cryptology, design theory, finite geometries, and scientific computation, among others. An important feature is the interplay between theory and applications which has led to many new perspectives in research on finite fields and other areas. This interplay has been emphasized in this series of conferences and certainly was reflected in Fq6. This volume offers up-to-date original research papers by leading experts in the area.
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Applications of Computational Algebraic Geometry

American Mathematical Society Short Course, January 6-7, 1997, San Diego, California

Author: Dinesh N. Manocha

Publisher: American Mathematical Soc.

ISBN: 0821807501

Category: Mathematics

Page: 172

View: 9762

This book introduces readers to key ideas and applications of computational algebraic geometry. Beginning with the discovery of Grobner bases and fueled by the advent of modern computers and the rediscovery of resultants, computational algebraic geometry has grown rapidly in importance. The fact that 'crunching equations' is now as easy as 'crunching numbers' has had a profound impact in recent years. At the same time, the mathematics used in computational algebraic geometry is unusually elegant and accessible, which makes the subject easy to learn and easy to apply. This book begins with an introduction to Grobner bases and resultants, then discusses some of the more recent methods for solving systems of polynomial equations. A sampler of possible applications follows, including computer-aided geometric design, complex information systems, integer programming, and algebraic coding theory. The lectures in the book assume no previous acquaintance with the material.
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Coding theory, design theory, group theory

proceedings of the Marshall Hall Conference

Author: Marshall Hall,Dieter Jungnickel,Scott A. Vanstone

Publisher: Wiley-Interscience

ISBN: N.A

Category: Mathematics

Page: 299

View: 4349

Contains papers prepared for the 1990 multidisciplinary conference held to honor the late mathematician and researcher. Topics include applications of classic geometry to finite geometries and designs; multiple transitive permutation groups; low dimensional groups and their geometry; difference sets in 2-groups; construction of Galois groups; construction of strongly p-imbeded subgroups in finite simple groups; Hall triple systems, Fisher spaces and 3-transposition groups; explicit embeddings in finitely generated groups; 2-transitive and flag transitive designs; efficient representations of perm groups; codes and combinatorial designs; optimal normal bases for finite fields; vector space designs from quadratic forms and inequalities; primitive permutation groups, graphs and relation algebras; large sets of ordered designs, orthogonal 1-factorizations and hyperovals; algebraic integers all of whose algebraic conjugates have the same absolute value.
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Geometric Algebra with Applications in Science and Engineering

Author: Eduardo Bayro Corrochano,Garret Sobczyk

Publisher: Springer Science & Business Media

ISBN: 9780817641993

Category: Computers

Page: 592

View: 4818

This book is addressed to a broad audience of cyberneticists, computer scientists, engineers, applied physicists and applied mathematicians. The book offers several examples to clarify the importance of geometric algebra in signal and image processing, filtering and neural computing, computer vision, robotics and geometric physics. The contributions of this book will help the reader to greater understand the potential of geometric algebra for the design and implementation of real time artifical systems.
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Algebraic K-Theory and Its Applications

Author: Jonathan Rosenberg

Publisher: Springer Science & Business Media

ISBN: 1461243149

Category: Mathematics

Page: 394

View: 3031

Algebraic K-Theory is crucial in many areas of modern mathematics, especially algebraic topology, number theory, algebraic geometry, and operator theory. This text is designed to help graduate students in other areas learn the basics of K-Theory and get a feel for its many applications. Topics include algebraic topology, homological algebra, algebraic number theory, and an introduction to cyclic homology and its interrelationship with K-Theory.
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Congressus Numerantium

Author: Ralph G. Stanton,Southeastern International Conference on Combinatorics, Graph Theory and Computing

Publisher: N.A

ISBN: 9780919628885

Category: Mathematics

Page: N.A

View: 3769

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Differential Geometry and Its Applications

Author: John Oprea

Publisher: MAA

ISBN: 9780883857489

Category: Mathematics

Page: 469

View: 1657

Differential geometry has a long, wonderful history it has found relevance in areas ranging from machinery design of the classification of four-manifolds to the creation of theories of nature's fundamental forces to the study of DNA. This book studies the differential geometry of surfaces with the goal of helping students make the transition from the compartmentalized courses in a standard university curriculum to a type of mathematics that is a unified whole, it mixes geometry, calculus, linear algebra, differential equations, complex variables, the calculus of variations, and notions from the sciences. Differential geometry is not just for mathematics majors, it is also for students in engineering and the sciences. Into the mix of these ideas comes the opportunity to visualize concepts through the use of computer algebra systems such as Maple. The book emphasizes that this visualization goes hand-in-hand with the understanding of the mathematics behind the computer construction. Students will not only “see” geodesics on surfaces, but they will also see the effect that an abstract result such as the Clairaut relation can have on geodesics. Furthermore, the book shows how the equations of motion of particles constrained to surfaces are actually types of geodesics. Students will also see how particles move under constraints. The book is rich in results and exercises that form a continuous spectrum, from those that depend on calculation to proofs that are quite abstract.
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Algebraic and Geometric Methods in Discrete Mathematics

Author: Heather A. Harrington,Mohamed Omar,Matthew Wright

Publisher: American Mathematical Soc.

ISBN: 1470423219

Category: Commutative algebra -- Computational aspects and applications -- Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.)

Page: 277

View: 3541

This volume contains the proceedings of the AMS Special Session on Algebraic and Geometric Methods in Applied Discrete Mathematics, held on January 11, 2015, in San Antonio, Texas. The papers present connections between techniques from “pure” mathematics and various applications amenable to the analysis of discrete models, encompassing applications of combinatorics, topology, algebra, geometry, optimization, and representation theory. Papers not only present novel results, but also survey the current state of knowledge of important topics in applied discrete mathematics. Particular highlights include: a new computational framework, based on geometric combinatorics, for structure prediction from RNA sequences; a new method for approximating the optimal solution of a sum of squares problem; a survey of recent Helly-type geometric theorems; applications of representation theory to voting theory and game theory; a study of fixed points of tensors; and exponential random graph models from the perspective of algebraic statistics with applications to networks. This volume was written for those trained in areas such as algebra, topology, geometry, and combinatorics who are interested in tackling problems in fields such as biology, the social sciences, data analysis, and optimization. It may be useful not only for experts, but also for students who wish to gain an applied or interdisciplinary perspective.
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Introduction to Coding Theory and Algebraic Geometry

Author: J. van Lint,G. van der Geer

Publisher: Birkhäuser

ISBN: 3034892861

Category: Juvenile Nonfiction

Page: 85

View: 4971

These notes are based on lectures given in the semmar on "Coding Theory and Algebraic Geometry" held at Schloss Mickeln, Diisseldorf, November 16-21, 1987. In 1982 Tsfasman, Vladut and Zink, using algebraic geometry and ideas of Goppa, constructed a seqeunce of codes that exceed the Gilbert-Varshamov bound. The result was considered sensational. Furthermore, it was surprising to see these unrelated areas of mathematics collaborating. The aim of this course is to give an introduction to coding theory and to sketch the ideas of algebraic geometry that led to the new result. Finally, a number of applications of these methods of algebraic geometry to coding theory are given. Since this is a new area, there are presently no references where one can find a more extensive treatment of all the material. However, both for algebraic geometry and for coding theory excellent textbooks are available. The combination ofthe two subjects can only be found in a number ofsurvey papers. A book by C. Moreno with a complete treatment of this area is in preparation. We hope that these notes will stimulate further research and collaboration of algebraic geometers and coding theorists. G. van der Geer, J.H. van Lint Introduction to CodingTheory and Algebraic Geometry PartI -- CodingTheory Jacobus H. vanLint 11 1. Finite fields In this chapter we collect (without proof) the facts from the theory of finite fields that we shall need in this course.
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