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Introduction to Coding Theory

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ISBN-10: 0521845041

ISBN-13: 9780521845045

Edition: 2005

Authors: Ron M. Roth

List price: $123.00
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Description:

Error-correcting codes constitute one of the key ingredients in achieving the high degree of reliability required in modern data transmission and storage systems. This book introduces the reader to the theoretical foundations of error-correcting codes, with an emphasis on Reed-Solomon codes and their derivative codes. After reviewing linear codes and finite fields, Ron Roth describes Reed-Solomon codes and various decoding algorithms. Cyclic codes are presented, as are MDS codes, graph codes, and codes in the Lee metric. Concatenated, trellis, and convolutional codes are also discussed in detail.
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Book details

List price: $123.00
Copyright year: 2005
Publisher: Cambridge University Press
Publication date: 2/23/2006
Binding: Hardcover
Pages: 580
Size: 7.25" wide x 9.75" long x 1.25" tall
Weight: 2.882
Language: English

Preface
Introduction
Communication systems
Channel coding
Block codes
Decoding
Levels of error handling
Problems
Notes
Linear Codes
Definition
Encoding of linear codes
Parity-check matrix
Decoding of linear codes
Problems
Notes
Introduction to Finite Fields
Prime fields
Polynomials
Extension fields
Roots of polynomials
Primitive elements
Field characteristic
Splitting field
Application: double error-correcting codes
Problems
Notes
Bounds on the Parameters of Codes
The Singleton bound
The sphere-packing bound
The Gilbert-Varshamov bound
MacWilliams' identities
Asymptotic bounds
Converse Coding Theorem
Coding Theorem
Problems
Notes
Reed-Solomon and Related Codes
Generalized Reed-Solomon codes
Conventional Reed-Solomon codes
Encoding of RS codes
Concatenated codes
Alternant codes
BCH codes
Problems
Notes
Decoding of Reed-Solomon Codes
Introduction
Syndrome computation
Key equation of GRS decoding
Solving the key equation by Euclid's algorithm
Finding the error values
Summary of the GRS decoding algorithm
The Berlekamp-Massey algorithm
Problems
Notes
Structure of Finite Fields
Minimal polynomials
Enumeration of irreducible polynomials
Isomorphism of finite fields
Primitive polynomials
Cyclotomic cosets
Problems
Notes
Cyclic Codes
Definition
Generator polynomial and check polynomial
Roots of a cyclic code
BCH codes as cyclic codes
The BCH bound
Problems
Notes
List Decoding of Reed-Solomon Codes
List decoding
Bivariate polynomials
GRS decoding through bivariate polynomials
Sudan's algorithm
The Guruswami-Sudan algorithm
List decoding of alternant codes
Fiding linear bivariate factors
Bounds on the decoding radius
Problems
Notes
Codes in the Lee Metric
Lee weight and Lee distance
Newton's identities
Lee-metric alternant codes and GRS codes
Decoding alternant codes in the Lee metric
Decoding GRS codes in the Lee metric
Berlekamp codes
Bounds for codes in the Lee metric
Problems
Notes
MDS Codes
Definition revisited
GRS codes and their extensions
Bounds on the length of linear MDS codes
GRS codes and the MDS conjecture
Uniqueness of certain MDS codes
Problems
Notes
Concatenated Codes
Definition revisited
Decoding of concatenated codes
The Zyablov bound
Justesen codes
Concatenated codes that attain capacity
Problems
Notes
Graph Codes
Basic concepts from graph theory
Regular graphs
Graph expansion
Expanders from codes
Ramanujan graphs
Codes from expanders
Iterative decoding of graph codes
Graph codes in concatenated schemes
Problems
Notes
Trellis and Convolutional Codes
Labeled directed graphs
Trellis codes
Decoding of trellis codes
Linear finite-state machines
Convolutional codes
Encoding of convolutional codes
Decoding of convolutional codes
Non-catastrophic generator matrices
Problems
Notes
Basics in Modern Algebra
Problems
Bibliography
List of Symbols
Index