Combinatorics And Finite Fields

Author : Kannan Soundararajan
Publisher : American Mathematical Society
Page : 100 pages
File Size : 40,8 Mb
Release : 2022-11-09
Category : Mathematics
ISBN : 9781470469306

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Finite Fields, with Applications to Combinatorics by Kannan Soundararajan Pdf

This book uses finite field theory as a hook to introduce the reader to a range of ideas from algebra and number theory. It constructs all finite fields from scratch and shows that they are unique up to isomorphism. As a payoff, several combinatorial applications of finite fields are given: Sidon sets and perfect difference sets, de Bruijn sequences and a magic trick of Persi Diaconis, and the polynomial time algorithm for primality testing due to Agrawal, Kayal and Saxena. The book forms the basis for a one term intensive course with students meeting weekly for multiple lectures and a discussion session. Readers can expect to develop familiarity with ideas in algebra (groups, rings and fields), and elementary number theory, which would help with later classes where these are developed in greater detail. And they will enjoy seeing the AKS primality test application tying together the many disparate topics from the book. The pre-requisites for reading this book are minimal: familiarity with proof writing, some linear algebra, and one variable calculus is assumed. This book is aimed at incoming undergraduate students with a strong interest in mathematics or computer science.

Author : Kai-Uwe Schmidt,Arne Winterhof
Publisher : Walter de Gruyter GmbH & Co KG
Page : 354 pages
File Size : 51,8 Mb
Release : 2019-07-08
Category : Mathematics
ISBN : 9783110642094

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Combinatorics and Finite Fields by Kai-Uwe Schmidt,Arne Winterhof Pdf

Combinatorics and finite fields are of great importance in modern applications such as in the analysis of algorithms, in information and communication theory, and in signal processing and coding theory. This book contains survey articles on topics such as difference sets, polynomials, and pseudorandomness.

Author : James A. Davis
Publisher : Walter de Gruyter GmbH & Co KG
Page : 214 pages
File Size : 41,8 Mb
Release : 2020-10-26
Category : Mathematics
ISBN : 9783110621730

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Finite Fields and their Applications by James A. Davis Pdf

The volume covers wide-ranging topics from Theory: structure of finite fields, normal bases, polynomials, function fields, APN functions. Computation: algorithms and complexity, polynomial factorization, decomposition and irreducibility testing, sequences and functions. Applications: algebraic coding theory, cryptography, algebraic geometry over finite fields, finite incidence geometry, designs, combinatorics, quantum information science.

Author : Gary L. Mullen,Daniel Panario
Publisher : CRC Press
Page : 1048 pages
File Size : 40,6 Mb
Release : 2013-06-17
Category : Computers
ISBN : 9781439873823

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Handbook of Finite Fields by Gary L. Mullen,Daniel Panario Pdf

Poised to become the leading reference in the field, the Handbook of Finite Fields is exclusively devoted to the theory and applications of finite fields. More than 80 international contributors compile state-of-the-art research in this definitive handbook. Edited by two renowned researchers, the book uses a uniform style and format throughout and

Author : Gary L. Mullen,Carl Mummert
Publisher : American Mathematical Soc.
Page : 175 pages
File Size : 44,5 Mb
Release : 2007
Category : Mathematics
ISBN : 9780821844182

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Finite Fields and Applications by Gary L. Mullen,Carl Mummert Pdf

Introduction to the theory of finite fields and to some of their many applications. The first chapter is devoted to the theory of finite fields. After covering their construction and elementary properties, the authors discuss the trace and norm functions, bases for finite fields, and properties of polynomials over finite fields. Chapter 2 deals with combinatorial topics such as the construction of sets of orthogonal Latin squares, affine and projective planes, block designs, and Hadamard matrices. Chapters 3 and 4 provide a number of constructions and basic properties of error-correcting codes and cryptographic systems using finite fields. Appendix A provides a brief review of the basic number theory and abstract algebra used in the text. Appendix B provides hints and partial solutions for many of the exercises in each chapter.--From publisher description.

Author : International Conference on Finite Fields and Applications
Publisher : American Mathematical Soc.
Page : 278 pages
File Size : 51,5 Mb
Release : 2008
Category : Coding theory
ISBN : 9780821843093

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Finite Fields and Applications by International Conference on Finite Fields and Applications Pdf

This volume contains the proceedings of the Eighth International Conference on Finite Fields and Applications, held in Melbourne, Australia, July 9-13, 2007. It contains 5 invited survey papers as well as original research articles covering various theoretical and applied areas related to finite fields.Finite fields, and the computational and algorithmic aspects of finite field problems, continue to grow in importance and interest in the mathematical and computer science communities because of their applications in so many diverse areas. In particular, finite fields now play very important roles in number theory, algebra, and algebraic geometry, as well as in computer science, statistics, and engineering. Areas of application include algebraic coding theory, cryptology, and combinatorialdesign theory.

Author : Simeon Ball
Publisher : Cambridge University Press
Page : 299 pages
File Size : 45,7 Mb
Release : 2015-07-02
Category : Mathematics
ISBN : 9781107107991

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Finite Geometry and Combinatorial Applications by Simeon Ball Pdf

A graduate-level introduction to finite geometry and its applications to other areas of combinatorics.

Author : Adriano M. Garsia,Ömer Eğecioğlu
Publisher : Springer Nature
Page : 243 pages
File Size : 50,8 Mb
Release : 2020-10-06
Category : Mathematics
ISBN : 9783030583736

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Lectures in Algebraic Combinatorics by Adriano M. Garsia,Ömer Eğecioğlu Pdf

Capturing Adriano Garsia's unique perspective on essential topics in algebraic combinatorics, this book consists of selected, classic notes on a number of topics based on lectures held at the University of California, San Diego over the past few decades. The topics presented share a common theme of describing interesting interplays between algebraic topics such as representation theory and elegant structures which are sometimes thought of as being outside the purview of classical combinatorics. The lectures reflect Garsia’s inimitable narrative style and his exceptional expository ability. The preface presents the historical viewpoint as well as Garsia's personal insights into the subject matter. The lectures then start with a clear treatment of Alfred Young's construction of the irreducible representations of the symmetric group, seminormal representations and Morphy elements. This is followed by an elegant application of SL(2) representations to algebraic combinatorics. The last two lectures are on heaps, continued fractions and orthogonal polynomials with applications, and finally there is an exposition on the theory of finite fields. The book is aimed at graduate students and researchers in the field.

Author : Pascale Charpin,Alexander Pott,Arne Winterhof
Publisher : Walter de Gruyter
Page : 288 pages
File Size : 52,6 Mb
Release : 2013-05-28
Category : Mathematics
ISBN : 9783110283600

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Finite Fields and Their Applications by Pascale Charpin,Alexander Pott,Arne Winterhof Pdf

This book is based on the invited talks of the "RICAM-Workshop on Finite Fields and Their Applications: Character Sums and Polynomials" held at the Federal Institute for Adult Education (BIfEB) in Strobl, Austria, from September 2-7, 2012. Finite fields play important roles in many application areas such as coding theory, cryptography, Monte Carlo and quasi-Monte Carlo methods, pseudorandom number generation, quantum computing, and wireless communication. In this book we will focus on sequences, character sums, and polynomials over finite fields in view of the above mentioned application areas: Chapters 1 and 2 deal with sequences mainly constructed via characters and analyzed using bounds on character sums. Chapters 3, 5, and 6 deal with polynomials over finite fields. Chapters 4 and 9 consider problems related to coding theory studied via finite geometry and additive combinatorics, respectively. Chapter 7 deals with quasirandom points in view of applications to numerical integration using quasi-Monte Carlo methods and simulation. Chapter 8 studies aspects of iterations of rational functions from which pseudorandom numbers for Monte Carlo methods can be derived. The goal of this book is giving an overview of several recent research directions as well as stimulating research in sequences and polynomials under the unified framework of character theory.

Author : James Hirschfeld,Joseph A. Thas
Publisher : Springer
Page : 409 pages
File Size : 50,6 Mb
Release : 2016-02-03
Category : Mathematics
ISBN : 9781447167907

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General Galois Geometries by James Hirschfeld,Joseph A. Thas Pdf

This book is the second edition of the third and last volume of a treatise on projective spaces over a finite field, also known as Galois geometries. This volume completes the trilogy comprised of plane case (first volume) and three dimensions (second volume). This revised edition includes much updating and new material. It is a mostly self-contained study of classical varieties over a finite field, related incidence structures and particular point sets in finite n-dimensional projective spaces. General Galois Geometries is suitable for PhD students and researchers in combinatorics and geometry. The separate chapters can be used for courses at postgraduate level.

Author : Dieter Jungnickel,H. Niederreiter
Publisher : Springer Science & Business Media
Page : 514 pages
File Size : 43,8 Mb
Release : 2001-03-20
Category : Mathematics
ISBN : 3540411097

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Finite Fields and Applications by Dieter Jungnickel,H. Niederreiter Pdf

This volume represents the refereed proceedings of the Fifth International Conference on Finite Fields and Applications (F q5) held at the University of Augsburg (Germany) from August 2-6, 1999, and hosted by the Department of Mathematics. The conference continued a series of biennial international conferences on finite fields, following earlier conferences at the University of Nevada at Las Vegas (USA) in August 1991 and August 1993, the University ofGlasgow (Scotland) in July 1995, and the University ofWaterloo (Canada) in August 1997. The Organizing Committee of F q5 comprised Thomas Beth (University ofKarlsruhe), Stephen D. Cohen (University of Glasgow), Dieter Jungnickel (University of Augsburg, Chairman), Alfred Menezes (University of Waterloo), Gary L. Mullen (Pennsylvania State University), Ronald C. Mullin (University of Waterloo), Harald Niederreiter (Austrian Academy of Sciences), and Alexander Pott (University of Magdeburg). The program ofthe conference consisted offour full days and one halfday ofsessions, with 11 invited plenary talks andover80contributedtalks that re- quired three parallel sessions. This documents the steadily increasing interest in finite fields and their applications. Finite fields have an inherently fasci- nating structure and they are important tools in discrete mathematics. Their applications range from combinatorial design theory, finite geometries, and algebraic geometry to coding theory, cryptology, and scientific computing. A particularly fruitful aspect is the interplay between theory and applications which has led to many new perspectives in research on finite fields.

Author : Gerald Berman,Kenneth D. Fryer
Publisher : Unknown
Page : 318 pages
File Size : 49,9 Mb
Release : 1972
Category : Mathematics
ISBN : UOM:39015040419510

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Introduction to Combinatorics by Gerald Berman,Kenneth D. Fryer Pdf

Author : Joel V. Brawley,George E. Schnibben
Publisher : American Mathematical Soc.
Page : 104 pages
File Size : 48,8 Mb
Release : 1989
Category : Mathematics
ISBN : 9780821851012

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Infinite Algebraic Extensions of Finite Fields by Joel V. Brawley,George E. Schnibben Pdf

Over the last several decades there has been a renewed interest in finite field theory, partly as a result of important applications in a number of diverse areas such as electronic communications, coding theory, combinatorics, designs, finite geometries, cryptography, and other portions of discrete mathematics. In addition, a number of recent books have been devoted to the subject. Despite the resurgence in interest, it is not widely known that many results concerning finite fields have natural generalizations to abritrary algebraic extensions of finite fields. The purpose of this book is to describe these generalizations. After an introductory chapter surveying pertinent results about finite fields, the book describes the lattice structure of fields between the finite field $GF(q)$ and its algebraic closure $\Gamma (q)$. The authors introduce a notion, due to Steinitz, of an extended positive integer $N$ which includes each ordinary positive integer $n$ as a special case. With the aid of these Steinitz numbers, the algebraic extensions of $GF(q)$ are represented by symbols of the form $GF(q^N)$. When $N$ is an ordinary integer $n$, this notation agrees with the usual notation $GF(q^n)$ for a dimension $n$ extension of $GF(q)$. The authors then show that many of the finite field results concerning $GF(q^n)$ are also true for $GF(q^N)$. One chapter is devoted to giving explicit algorithms for computing in several of the infinite fields $GF(q^N)$ using the notion of an explicit basis for $GF(q^N)$ over $GF(q)$. Another chapter considers polynomials and polynomial-like functions on $GF(q^N)$ and contains a description of several classes of permutation polynomials, including the $q$-polynomials and the Dickson polynomials. Also included is a brief chapter describing two of many potential applications. Aimed at the level of a beginning graduate student or advanced undergraduate, this book could serve well as a supplementary text for a course in finite field theory.

Author : Dieter Jungnickel,H. Niederreiter
Publisher : Springer
Page : 490 pages
File Size : 53,6 Mb
Release : 2012-10-23
Category : Mathematics
ISBN : 3642624987

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Finite Fields and Applications by Dieter Jungnickel,H. Niederreiter Pdf

This volume represents the refereed proceedings of the Fifth International Conference on Finite Fields and Applications (F q5) held at the University of Augsburg (Germany) from August 2-6, 1999, and hosted by the Department of Mathematics. The conference continued a series of biennial international conferences on finite fields, following earlier conferences at the University of Nevada at Las Vegas (USA) in August 1991 and August 1993, the University ofGlasgow (Scotland) in July 1995, and the University ofWaterloo (Canada) in August 1997. The Organizing Committee of F q5 comprised Thomas Beth (University ofKarlsruhe), Stephen D. Cohen (University of Glasgow), Dieter Jungnickel (University of Augsburg, Chairman), Alfred Menezes (University of Waterloo), Gary L. Mullen (Pennsylvania State University), Ronald C. Mullin (University of Waterloo), Harald Niederreiter (Austrian Academy of Sciences), and Alexander Pott (University of Magdeburg). The program ofthe conference consisted offour full days and one halfday ofsessions, with 11 invited plenary talks andover80contributedtalks that re quired three parallel sessions. This documents the steadily increasing interest in finite fields and their applications. Finite fields have an inherently fasci nating structure and they are important tools in discrete mathematics. Their applications range from combinatorial design theory, finite geometries, and algebraic geometry to coding theory, cryptology, and scientific computing. A particularly fruitful aspect is the interplay between theory and applications which has led to many new perspectives in research on finite fields.

Author : Dirk Hachenberger,Dieter Jungnickel
Publisher : Springer Nature
Page : 785 pages
File Size : 55,9 Mb
Release : 2020-09-29
Category : Mathematics
ISBN : 9783030608064

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Topics in Galois Fields by Dirk Hachenberger,Dieter Jungnickel Pdf

This monograph provides a self-contained presentation of the foundations of finite fields, including a detailed treatment of their algebraic closures. It also covers important advanced topics which are not yet found in textbooks: the primitive normal basis theorem, the existence of primitive elements in affine hyperplanes, and the Niederreiter method for factoring polynomials over finite fields. We give streamlined and/or clearer proofs for many fundamental results and treat some classical material in an innovative manner. In particular, we emphasize the interplay between arithmetical and structural results, and we introduce Berlekamp algebras in a novel way which provides a deeper understanding of Berlekamp's celebrated factorization algorithm. The book provides a thorough grounding in finite field theory for graduate students and researchers in mathematics. In view of its emphasis on applicable and computational aspects, it is also useful for readers working in information and communication engineering, for instance, in signal processing, coding theory, cryptography or computer science.