*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:** 7001

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.

'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.

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).

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

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.

This volume covers many topics, including number theory, Boolean functions, combinatorial geometry, and algorithms over finite fields. It contains many new, theoretical and applicable results, as well as surveys that were presented by the top specialists in these areas. New results include an answer to one of Serre''s questions, posted in a letter to Top; cryptographic applications of the discrete logarithm problem related to elliptic curves and hyperelliptic curves; construction of function field towers; construction of new classes of Boolean cryptographic functions; and algorithmic applications of algebraic geometry. Sample Chapter(s). Chapter 1: Fast addition on non-hyperelliptic genus 3 curves (424 KB). Contents: Symmetric Cryptography and Algebraic Curves (F Voloch); Galois Invariant Smoothness Basis (J-M Couveignes & R Lercier); Fuzzy Pairing-Based CL-PKC (M Kiviharju); On the Semiprimitivity of Cyclic Codes (Y Aubry & P Langevin); Decoding of Scroll Codes (G H Hitching & T Johnsen); An Optimal Unramified Tower of Function Fields (K Brander); On the Number of Resilient Boolean Functions (S Mesnager); On Quadratic Extensions of Cyclic Projective Planes (H F Law & P P W Wong); Partitions of Vector Spaces over Finite Fields (Y Zelenyuk); and other papers. Readership: Mathematicians, researchers in mathematics (academic and industry R&D).

Proceedings of the NATO Advanced Research Workshop, held in Eilat, Israel, from 25th February to 1st March 2001

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.

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.

*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:** 1953

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.

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.

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.

This book connects coding theory with actual applications in consumer electronics and with other areas of mathematics. ""Different Aspects of Coding Theory"" covers in detail the mathematical foundations of digital data storage and makes connections to symbolic dynamics, linear systems, and finite automata. It also explores the use of algebraic geometry within coding theory and examines links with finite geometry, statistics, and theoretical computer science. This book features: a unique combination of mathematical theory and engineering practice; much diversity and variety among chapters, thus offering broad appeal; and, topics relevant to mathematicians, statisticians, engineers, and computer scientists. Contributions are by recognized scholars.

**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:** 1967