Information and Coding Theory

By: Gareth A. Jones, J.Mary Jones

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Paperback | 1 July 2000

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228 Pages

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23.5 x 17.15 x 1.91

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This book provides an elementary introduction to Information Theory and Coding Theory - two related aspects of the problem of how to transmit information efficiently and accurately. The first part of the book focuses on Information Theory, covering uniquely decodable and instantaneous codes, Huffman coding, entropy, information channels, and Shannon's Fundamental Theorem. In the second part, on Coding Theory, linear algebra is used to construct examples of such codes, such as the Hamming, Hadamard, Golay and Reed-Muller codes.The book emphasises carefully explained proofs and worked examples; exercises (with solutions) are integrated into the text as part of the learning process. Only some basic probability theory and linear algebra, together with a little calculus (as covered in most first-year university syllabuses), is assumed, making it suitable for second- and third-year undergraduates in mathematics, electronics and computer science.

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You Can Find This Book In

Non-Fiction Reference, Information & Interdisciplinary Subjects Research & Information Information theory Engineering & Technology Technology in General Maths for Engineers Coding Theory & Cryptology Science Science in General Maths for Scientists Mathematics

Applied Mathematics Computing & I.T.Computer Science Systems Analysis & Design Probability & Statistics Combinatorics & Graph Theory Number Theory

Preface	p. v
Notes to the Reader	p. xiii
Source Coding	p. 1
Definitions and Examples	p. 1
Uniquely Decodable Codes	p. 4
Instantaneous Codes	p. 9
Constructing Instantaneous Codes	p. 11
Kraft's Inequality	p. 13
McMillan's Inequality	p. 14
Comments on Kraft's and McMillan's Inequalities	p. 16
Supplementary Exercises	p. 17
Optimal Codes	p. 19
Optimality	p. 19
Binary Huffman Codes	p. 22
Average Word-length of Huffman Codes	p. 26
Optimality of Binary Huffman Codes	p. 27
r-ary Huffman Codes	p. 28
Extensions of Sources	p. 30
Supplementary Exercises	p. 32
Entropy	p. 35
Information and Entropy	p. 35
Properties of the Entropy Function	p. 40
Entropy and Average Word-length	p. 42
Shannon-Fano Coding	p. 45
Entropy of Extensions and Products	p. 47
Shannon's First Theorem	p. 48
An Example of Shannon's First Theorem	p. 49
Supplementary Exercises	p. 51
Information Channels	p. 55
Notation and Definitions	p. 55
The Binary Symmetric Channel	p. 60
System Entropies	p. 62
System Entropies for the Binary Symmetric Channel	p. 64
Extension of Shannon's First Theorem to Information Channels	p. 67
Mutual Information	p. 70
Mutual Information for the Binary Symmetric Channel	p. 72
Channel Capacity	p. 73
Supplementary Exercises	p. 76
Using an Unreliable Channel	p. 79
Decision Rules	p. 79
An Example of Improved Reliability	p. 82
Hamming Distance	p. 85
Statement and Outline Proof of Shannon's Theorem	p. 88
The Converse of Shannon's Theorem	p. 90
Comments on Shannon's Theorem	p. 93
Supplementary Exercises	p. 94
Error-correcting Codes	p. 97
Introductory Concepts	p. 97
Examples of Codes	p. 100
Minimum Distance	p. 104
Hamming's Sphere-packing Bound	p. 107
The Gilbert-Varshamov Bound	p. 111
Hadamard Matrices and Codes	p. 114
Supplementary Exercises	p. 118
Linear Codes	p. 121
Matrix Description of Linear Codes	p. 121
Equivalence of Linear Codes	p. 127
Minimum Distance of Linear Codes	p. 131
The Hamming Codes	p. 133
The Golay Codes	p. 136
The Standard Array	p. 141
Syndrome Decoding	p. 143
Supplementary Exercises	p. 146
Suggestions for Further Reading	p. 149
Proof of the Sardinas-Patterson Theorem	p. 153
The Law of Large Numbers	p. 157
Proof of Shannon's Fundamental Theorem	p. 159
Solutions to Exercises	p. 165
Bibliography	p. 191
Index of Symbols and Abbreviations	p. 195
Index	p. 201
Table of Contents provided by Syndetics. All Rights Reserved.

Information and Coding Theory

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