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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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As this Preface is being written, the twentieth century is coming to an end. Historians may perhaps come to refer to it as the century of information, just as its predecessor is associated with the process of industrialisation. Successive technological developments such as the telephone, radio, television, computers and the Internet have had profound effects on the way we live. We can see pic tures of the surface of Mars or the early shape of the Universe. The contents of a whole shelf-load of library books can be compressed onto an almost weight less piece of plastic. Billions of people can watch the same football match, or can keep in instant touch with friends around the world without leaving home. In short, massive amounts of information can now be stored, transmitted and processed, with surprising speed, accuracy and economy. Of course, these developments do not happen without some theoretical ba sis, and as is so often the case, much of this is provided by mathematics. Many of the first mathematical advances in this area were made in the mid-twentieth century by engineers, often relying on intuition and experience rather than a deep theoretical knowledge to lead them to their discoveries. Soon the math ematicians, delighted to see new applications for their subject, joined in and developed the engineers' practical examples into wide-ranging theories, com plete with definitions, theorems and proofs.

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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
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Information and Coding Theory

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