# Book:Robert B. Ash/Information Theory

## Robert B. Ash: Information Theory

Published $\text {1965}$, Dover Publications

ISBN 0-486-66521-6.

### Contents

Preface
Chapter One: A Measure of Information
1.1 Introduction
1.2 Axioms for the Uncertainty Measure
1.3 Three Interpretations of the Uncertainty Function
1.4 Properties of the Uncertainty Function; Joint and Conditional Uncertainty
1.5 The Measure of Information
1.6 Notes and Remarks
Chapter Two: Noiseless Coding
2.1 Introduction
2.2 The Problem of Unique Decipherability
2.3 Necessary and Sufficient Conditions for the Existence of Instantaneous Codes
2.4 Extension of the Condition $\displaystyle \sum_{i \mathop = 1}^M D^{-n_i} \le 1$ to Uniquely Decipherable Codes
2.5 The Noiseless Coding Theorem
2.6 Construction of Optimal Codes
2.7 Notes and Remarks
Chapter Three: The Discrete Memoryless Channel
3.1 Models for Communication Channels
3.2 The Information Processed by a Channel; Channel Capacity; Classification of Channels
3.3 Calculation of Channel Capacity
3.4 Decoding Schemes; the Ideal Observer
3.5 The Fundamental Theorem
3.6 Exponential Error Bounds
3.7 The Weak Converse to the Fundamental Theorem
3.8 Notes and Remarks
Chapter Four: Error Correcting Codes
4.1 Introduction; Minimum Distance Principle
4.2 Relation between Distance and Error Correcting Properties of Codes; the Hamming Bound
4.3 Parity Check Coding
4.4 The Application of Group Theory to Parity Check Coding
4.5 Upper and Lower Bounds on the Error Correcting Ability of Parity Check Codes
4.6 Parity Check Codes Are Adequate
4.7 Precise Error Bounds for General Binary Codes
4.8 The Strong Converse for the Binary Symmetric Channel
4.9 Non-Binary Coding
4.10 Notes and Remarks
Chapter Five: Further Theory of Error Correcting Codes
5.1 Feedback Shift Registers and Cyclic Codes
5.2 General Properties of Binary Matrices and Their Cycle Sets
5.3 Properties of Cyclic Codes
5.4 Bose-Chaudhuri-Hocquenghem Codes
5.5 Single Error Correcting Cyclic Codes; Automatic Decoding
5.6 Notes and Remarks
Chapter Six: Information Sources
6.1 Introduction
6.2 A Mathematical Model for an Information Source
6.3 Introduction to the Theory of Finite Markov Groups
6.4 Information Sources; Uncertainty of a Source
6.5 Order of a Source; Approximation of a General Information Source by a Source of Finite Order
6.6 The Asymptotic Equipartition Property
6.7 Notes and Remarks
Chapter Seven: Channels with Memory
7.1 Introduction
7.2 The Finite-State Channel
7.3 The Coding Theorem for Finite State Regular Channels
7.4 The Capacity of a General Discrete Channel; Comparison of the Weak and Strong Converses
7.5 Notes and Remarks
Chapter Eight: Continuous Channels
8.1 Introduction
8.2 The Time-Discrete Gaussian Channel
8.3 Uncertainty in the Continuous Case
8.4 The Converse to the Coding Theorem for the Time-Discrete Gaussian Channel
8.5 The Time-Continuous Gaussian Channel
8.6 Band-Limited Channels
8.7 Notes and Remarks
Appendix
1. Compact and Symmetric Operators on $L_2 \sqbrk {a, b}$
2. Integral Operators
3. The Karhunen-Loève Theorem
4. Further Results Concerning Integral Operators Determined by a Covariance Function
Tables of Values of $-\log_2 p$ and $-p \log_2 p$
Solutions to Problems
References
Index