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Codes From Difference Sets

Author : Cunsheng Ding

Publisher : World Scientific

Page : 354 pages

File Size : 41,58 MB

Release : 2014-08-08

Category : Mathematics

ISBN : 981461937X

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Book Description

This is the first monograph on codebooks and linear codes from difference sets and almost difference sets. It aims at providing a survey of constructions of difference sets and almost difference sets as well as an in-depth treatment of codebooks and linear codes from difference sets and almost difference sets. To be self-contained, this monograph covers necessary mathematical foundations and the basics of coding theory. It also contains tables of best BCH codes and best cyclic codes over GF(2) and GF(3) up to length 125 and 79, respectively. This repository of tables can be used to benchmark newly constructed cyclic codes. This monograph is intended to be a reference for postgraduates and researchers who work on combinatorics, or coding theory, or digital communications.



Difference Sets

Author : Emily H. Moore

Publisher : American Mathematical Soc.

Page : 315 pages

File Size : 25,68 MB

Release : 2013-06-13

Category : Mathematics

ISBN : 0821891766

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Book Description

Difference sets belong both to group theory and to combinatorics. Studying them requires tools from geometry, number theory, and representation theory. This book lays a foundation for these topics, including a primer on representations and characters of f



Difference Sets, Sequences and their Correlation Properties

Author : A. Pott

Publisher : Springer Science & Business Media

Page : 451 pages

File Size : 26,42 MB

Release : 2012-12-06

Category : Mathematics

ISBN : 9401144591

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Book Description

The explanation of the formal duality of Kerdock and Preparata codes is one of the outstanding results in the field of applied algebra in the last few years. This result is related to the discovery of large sets of quad riphase sequences over Z4 whose correlation properties are better than those of the best binary sequences. Moreover, the correlation properties of sequences are closely related to difference properties of certain sets in (cyclic) groups. It is the purpose of this book to illustrate the connection between these three topics. Most articles grew out of lectures given at the NATO Ad vanced Study Institute on "Difference sets, sequences and their correlation properties". This workshop took place in Bad Windsheim (Germany) in August 1998. The editors thank the NATO Scientific Affairs Division for the generous support of this workshop. Without this support, the present collection of articles would not have been realized.



Contemporary Design Theory

Author : Jeffrey H. Dinitz

Publisher : John Wiley & Sons

Page : 660 pages

File Size : 27,95 MB

Release : 1992-08-04

Category : Mathematics

ISBN : 9780471531418

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Book Description

Foremost experts in their field have contributed articles resulting in a compilation of useful and timely surveys in this ever-expanding field. Each of these 12 original papers covers important aspects of design theory including several in areas that have not previously been surveyed. Also contains surveys updating earlier ones where research is particularly active.



Concise Encyclopedia of Coding Theory

Author : W. Cary Huffman

Publisher : CRC Press

Page : 998 pages

File Size : 28,91 MB

Release : 2021-03-26

Category : Computers

ISBN : 1351375105

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Book Description

Most coding theory experts date the origin of the subject with the 1948 publication of A Mathematical Theory of Communication by Claude Shannon. Since then, coding theory has grown into a discipline with many practical applications (antennas, networks, memories), requiring various mathematical techniques, from commutative algebra, to semi-definite programming, to algebraic geometry. Most topics covered in the Concise Encyclopedia of Coding Theory are presented in short sections at an introductory level and progress from basic to advanced level, with definitions, examples, and many references. The book is divided into three parts: Part I fundamentals: cyclic codes, skew cyclic codes, quasi-cyclic codes, self-dual codes, codes and designs, codes over rings, convolutional codes, performance bounds Part II families: AG codes, group algebra codes, few-weight codes, Boolean function codes, codes over graphs Part III applications: alternative metrics, algorithmic techniques, interpolation decoding, pseudo-random sequences, lattices, quantum coding, space-time codes, network coding, distributed storage, secret-sharing, and code-based-cryptography. Features Suitable for students and researchers in a wide range of mathematical disciplines Contains many examples and references Most topics take the reader to the frontiers of research



Designs and Their Codes

Author : E. F. Assmus

Publisher : Cambridge University Press

Page : 366 pages

File Size : 50,77 MB

Release : 1994-01-06

Category : Mathematics

ISBN : 9780521458399

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Book Description

A self-contained account suited for a wide audience describing coding theory, combinatorial designs and their relations.



Combinatorial Designs and their Applications

Author : Kathleen Quinn

Publisher : CRC Press

Page : 164 pages

File Size : 18,28 MB

Release : 1999-01-29

Category : Mathematics

ISBN : 9780849306594

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Book Description

The fruit of a conference that gathered seven very active researchers in the field, Combinatorial Design and their Applications presents a wide but representative range of topics on the non-geometrical aspects of design theory. By concentrating on a few important areas, the authors succeed in providing greater detail in these areas in a more complete and accessible form. Through their contributions to this collection, they help fill a gap in the available combinatorics literature. The papers included in this volume cover recent developments in areas of current interest, such as difference sets, cryptography, and optimal linear codes. Researchers in combinatorics and other areas of pure mathematics, along with researchers in statistics and computer design will find in-depth, up-to-date discussions of design theory and the application of the theory to statistical design, codes, and cryptography.



An Introduction to Algebraic and Combinatorial Coding Theory

Author : Ian F. Blake

Publisher : Academic Press

Page : 244 pages

File Size : 34,55 MB

Release : 2014-05-10

Category : Mathematics

ISBN : 1483260291

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Book Description

An Introduction to Algebraic and Combinatorial Coding Theory focuses on the principles, operations, and approaches involved in the combinatorial coding theory, including linear transformations, chain groups, vector spaces, and combinatorial constructions. The publication first offers information on finite fields and coding theory and combinatorial constructions and coding. Discussions focus on quadratic residues and codes, self-dual and quasicyclic codes, balanced incomplete block designs and codes, polynomial approach to coding, and linear transformations of vector spaces over finite fields. The text then examines coding and combinatorics, including chains and chain groups, equidistant codes, matroids, graphs, and coding, matroids, and dual chain groups. The manuscript also ponders on Möbius inversion formula, Lucas's theorem, and Mathieu groups. The publication is a valuable source of information for mathematicians and researchers interested in the combinatorial coding theory.



The Mathematical Theory of Coding

Author : Ian F. Blake

Publisher : Academic Press

Page : 369 pages

File Size : 15,55 MB

Release : 2014-05-10

Category : Mathematics

ISBN : 1483260593

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Book Description

The Mathematical Theory of Coding focuses on the application of algebraic and combinatoric methods to the coding theory, including linear transformations, vector spaces, and combinatorics. The publication first offers information on finite fields and coding theory and combinatorial constructions and coding. Discussions focus on self-dual and quasicyclic codes, quadratic residues and codes, balanced incomplete block designs and codes, bounds on code dictionaries, code invariance under permutation groups, and linear transformations of vector spaces over finite fields. The text then takes a look at coding and combinatorics and the structure of semisimple rings. Topics include structure of cyclic codes and semisimple rings, group algebra and group characters, rings, ideals, and the minimum condition, chains and chain groups, dual chain groups, and matroids, graphs, and coding. The book ponders on group representations and group codes for the Gaussian channel, including distance properties of group codes, initial vector problem, modules, group algebras, andrepresentations, orthogonality relationships and properties of group characters, and representation of groups. The manuscript is a valuable source of data for mathematicians and researchers interested in the mathematical theory of coding.



Algebraic Design Theory and Hadamard Matrices

Author : Charles J. Colbourn

Publisher : Springer

Page : 261 pages

File Size : 44,75 MB

Release : 2015-09-03

Category : Mathematics

ISBN : 331917729X

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Book Description

This volume develops the depth and breadth of the mathematics underlying the construction and analysis of Hadamard matrices, and their use in the construction of combinatorial designs. At the same time, it pursues current research in their numerous applications in security and cryptography, quantum information, and communications. Bridges among diverse mathematical threads and extensive applications make this an invaluable source for understanding both the current state of the art and future directions.​ ​The existence of Hadamard matrices remains one of the most challenging open questions in combinatorics. Substantial progress on their existence has resulted from advances in algebraic design theory using deep connections with linear algebra, abstract algebra, finite geometry, number theory, and combinatorics. Hadamard matrices arise in a very diverse set of applications. Starting with applications in experimental design theory and the theory of error-correcting codes, they have found unexpected and important applications in cryptography, quantum information theory, communications, and networking.