Ronald C. Mullin – författare

The theory of finite fields, whose origins can be traced back to the works of Gauss and Galois, has played a part in various branches in mathematics. Inrecent years we have witnessed a resurgence of interest in finite fields, and this is partly due to important applications in coding theory and cryptography. The purpose of this book is to introduce the reader to some of these recent developments. It should be of interest to a wide range of students, researchers and practitioners in the disciplines of computer science, engineering and mathematics. We shall focus our attention on some specific recent developments in the theory and applications of finite fields. While the topics selected are treated in some depth, we have not attempted to be encyclopedic. Among the topics studied are different methods of representing the elements of a finite field (including normal bases and optimal normal bases), algorithms for factoring polynomials over finite fields, methods for constructing irreducible polynomials, the discrete logarithm problem and its implications to cryptography, the use of elliptic curves in constructing public key cryptosystems, and the uses of algebraic geometry in constructing good error-correcting codes. To limit the size of the volume we have been forced to omit some important applications of finite fields. Some of these missing applications are briefly mentioned in the Appendix along with some key references.

50 Years of Combinatorics, Graph Theory, and Computing advances research in discrete mathematics by providing current research surveys, each written by experts in their subjects.

The book also celebrates outstanding mathematics from 50 years at the Southeastern International Conference on Combinatorics, Graph Theory & Computing (SEICCGTC). The conference is noted for the dissemination and stimulation of research, while fostering collaborations among mathematical scientists at all stages of their careers.

The authors of the chapters highlight open questions. The sections of the book include: Combinatorics; Graph Theory; Combinatorial Matrix Theory; Designs, Geometry, Packing and Covering. Readers will discover the breadth and depth of the presentations at the SEICCGTC, as well as current research in combinatorics, graph theory and computer science.

Features:

Commemorates 50 years of the Southeastern International Conference on Combinatorics, Graph Theory & Computing with research surveys

Surveys highlight open questions to inspire further research

Chapters are written by experts in their fields

Extensive bibliographies are provided at the end of each chapter

50 Years of Combinatorics, Graph Theory, and Computing advances research in discrete mathematics by providing current research surveys, each written by experts in their subjects.

The book also celebrates outstanding mathematics from 50 years at the Southeastern International Conference on Combinatorics, Graph Theory & Computing (SEICCGTC). The conference is noted for the dissemination and stimulation of research, while fostering collaborations among mathematical scientists at all stages of their careers.

The authors of the chapters highlight open questions. The sections of the book include: Combinatorics; Graph Theory; Combinatorial Matrix Theory; Designs, Geometry, Packing and Covering. Readers will discover the breadth and depth of the presentations at the SEICCGTC, as well as current research in combinatorics, graph theory and computer science.

Features:

Commemorates 50 years of the Southeastern International Conference on Combinatorics, Graph Theory & Computing with research surveys

Surveys highlight open questions to inspire further research

Chapters are written by experts in their fields

Extensive bibliographies are provided at the end of each chapter

50 Years of Combinatorics, Graph Theory, and Computing advances research in discrete mathematics by providing current research surveys, each written by experts in their subjects. The book also celebrates outstanding mathematics from 50 years at the Southeastern International Conference on Combinatorics, Graph Theory & Computing (SEICCGTC). The conference is noted for the dissemination and stimulation of research, while fostering collaborations among mathematical scientists at all stages of their careers. The authors of the chapters highlight open questions. The sections of the book include: Combinatorics; Graph Theory; Combinatorial Matrix Theory; Designs, Geometry, Packing and Covering. Readers will discover the breadth and depth of the presentations at the SEICCGTC, as well as current research in combinatorics, graph theory and computer science.Features:Commemorates 50 years of the Southeastern International Conference on Combinatorics, Graph Theory & Computing with research surveysSurveys highlight open questions to inspire further researchChapters are written by experts in their fieldsExtensive bibliographies are provided at the end of each chapter

The theory of finite fields, whose origins can be traced back to the works of Gauss and Galois, has played a part in various branches in mathematics. Inrecent years we have witnessed a resurgence of interest in finite fields, and this is partly due to important applications in coding theory and cryptography. The purpose of this book is to introduce the reader to some of these recent developments. It should be of interest to a wide range of students, researchers and practitioners in the disciplines of computer science, engineering and mathematics. We shall focus our attention on some specific recent developments in the theory and applications of finite fields. While the topics selected are treated in some depth, we have not attempted to be encyclopedic. Among the topics studied are different methods of representing the elements of a finite field (including normal bases and optimal normal bases), algorithms for factoring polynomials over finite fields, methods for constructing irreducible polynomials, the discrete logarithm problem and its implications to cryptography, the use of elliptic curves in constructing public key cryptosystems, and the uses of algebraic geometry in constructing good error-correcting codes. To limit the size of the volume we have been forced to omit some important applications of finite fields. Some of these missing applications are briefly mentioned in the Appendix along with some key references.

The theory of finite fields, whose origins can be traced back to the works of Gauss and Galois, has played a part in various branches in mathematics. Inrecent years we have witnessed a resurgence of interest in finite fields, and this is partly due to important applications in coding theory and cryptography. The purpose of this book is to introduce the reader to some of these recent developments. It should be of interest to a wide range of students, researchers and practitioners in the disciplines of computer science, engineering and mathematics. We shall focus our attention on some specific recent developments in the theory and applications of finite fields. While the topics selected are treated in some depth, we have not attempted to be encyclopedic. Among the topics studied are different methods of representing the elements of a finite field (including normal bases and optimal normal bases), algorithms for factoring polynomials over finite fields, methods for constructing irreducible polynomials, the discrete logarithm problem and its implications to cryptography, the use of elliptic curves in constructing public key cryptosystems, and the uses of algebraic geometry in constructing good error-correcting codes. To limit the size of the volume we have been forced to omit some important applications of finite fields. Some of these missing applications are briefly mentioned in the Appendix along with some key references.

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.