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Finite Commutative Rings and Their Applications

By: Gilberto Bini, Flaminio Flamini

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Hardcover | 30 April 2002

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Hardcover
192 Pages

Dimensions(cm)
24.13 x 16.51 x 1.91

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Foreword by Dieter Jungnickel
Finite Commutative Rings and their Applications answers a need for an introductory reference in finite commutative ring theory as applied to information and communication theory. This book will be of interest to both professional and academic researchers in the fields of communication and coding theory.

The book is a concrete and self-contained introduction to finite commutative local rings, focusing in particular on Galois and Quasi-Galois rings. The reader is provided with an active and concrete approach to the study of the purely algebraic structure and properties of finite commutative rings (in particular, Galois rings) as well as to their applications to coding theory.

Finite Commutative Rings and their Applications is the first to address both theoretical and practical aspects of finite ring theory. The authors provide a practical approach to finite rings through explanatory examples, thereby avoiding an abstract presentation of the subject. The section on Quasi-Galois rings presents new and unpublished results as well. The authors then introduce some applications of finite rings, in particular Galois rings, to coding theory, using a solid algebraic and geometric theoretical background.

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You Can Find This Book In

Non-Fiction Mathematics Algebra Science Science in General Medicine Medicine in General Reference, Information & Interdisciplinary Subjects Research & Information Information theory Cybernetics & Systems Theory Groups & Group Theory

Engineering & Technology Energy Technology & Engineering Electrical Engineering Coding Theory & Cryptology Discrete Mathematics

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Paperback

Published: 26th October 2012

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Foreword	p. vii
Preface	p. ix
Fundamental Notions in Ring Theory	p. 1
Basic Definitions	p. 1
Prime and Maximal Ideals	p. 3
Euclidean Domains, P.I.D.'s and U.F.D.'s	p. 9
Factorization in Z[subscript p[superscript n] [x]	p. 20
Finite Field Structure	p. 29
Basic Properties	p. 29
Characterization of Finite Fields	p. 31
Galois Field Automorphisms	p. 34
Finite Commutative Rings. Regular Polynomials	p. 39
Finite Commutative Ring Structure	p. 39
Regular Polynomials in the Ring R[x]	p. 47
R-algebra Automorphisms of R[x]	p. 54
Factorization in R[x]	p. 56
Separable Extensions of Finite Fields and Finite Rings	p. 61
Separable Field Extensions	p. 61
Extensions of Rings	p. 65
Separable extensions of finite commutative local rings	p. 67
Galois Theory for Local Rings	p. 71
Basic Facts	p. 71
Examples. Splitting Rings	p. 75
Galois and Quasi-Galois Rings: Structure and Properties	p. 81
Classical Constructions	p. 82
Galois Ring Properties	p. 92
Structure Theorems for finite commutative local rings	p. 105
Another class of finite commutative local rings: Quasi-Galois Rings	p. 107
Basic Notions on Codes Over Finite Fields	p. 121
Basic properties	p. 121
Some families of q-ary codes	p. 122
Linear Codes	p. 122
Hamming codes	p. 123
Cyclic codes	p. 124
Reed-Muller codes	p. 129
Duality between codes	p. 130
Some families of nonlinear q-ary codes	p. 134
Binary Kerdock codes	p. 135
Kerdock sets	p. 135
Properties of binary Kerdock codes	p. 138
Classical Preparata codes	p. 140
Basic properties	p. 141
Preparata codes and Hamming codes	p. 142
Basic Notions on Codes Over Galois Rings	p. 145
Basic properties	p. 145
Linear codes over Z[subscript p[superscript n]	p. 146
Reed-Muller codes over Z[subscript p[superscript n]	p. 147
Cyclic codes over Z[subscript p[superscript n]	p. 148
Hamming codes over Z[subscript p[superscript n]	p. 151
Linear quaternary codes	p. 152
Kerdock and Preparata codes revisited	p. 158
Bibliography	p. 167
Index	p. 173
Table of Contents provided by Syndetics. All Rights Reserved.

Finite Commutative Rings and Their Applications

At a Glance

Hardcover

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