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Tables of Contents for A Commonsense Approach to the Theory of Error Correcting Codes
Chapter/Section Title
Page #
Page Count
Preface
ix
 
Introduction
1
12
A Communication System and the Effect of Errors
1
2
The Concept of Parity
3
2
Error Detection
5
4
The Concept of Block Codes
9
1
Review of the Concepts Introduced
10
3
Linear Codes
13
30
Basic Concepts
13
2
Minimum Hamming Distance of a Code and the Error Detection/Correction Capability of the Code
15
5
Hamming Codes
20
7
The Parity Matrix of a Linear Code
27
7
Constructing a General Hamming Code
34
2
The Generating Matrix of a Systematic Code
36
5
Review of the Concepts Introduced
41
2
Basic Circuitry
43
22
Automatic Generation of the Rows of the Generating Matrix
43
4
Multiplying a Message by the Parity Matrix
47
3
The Encoding Process
50
4
Encoding/Decoding of Long and Short Vectors
54
3
Automatic Single Error Correction
57
3
A General Single Error Correcting Code
60
5
Cyclic Codes
65
38
Some Properties of Matrices Generated by LFSR
65
5
Linear Shifts of Code Words and Their Effect on the Generating Matrix
70
4
Properties of Maximum-Length Code Words
74
4
The Encoder and Decoder of a Code Satisfying the Parity Equations of Several Independent Codes
78
7
Maximum-Length Sequences
85
18
Single Burst Error Correction
103
44
The Definition of a Burst Error
103
2
Single Burst Error Detection
105
4
The Connection between the Burst Pattern and Cyclic Shifts of the Error Syndrome
109
2
Connection between the Location of a Burst Error and the Amount of Cyclic Shift between the Burst Pattern and the Error Syndrome
111
3
Conditions under Which It Is Possible to Detect the Exact Burst Pattern
114
2
Ways of Recovering the Location of a Burst Error
116
7
Burst Error Correction Circuitry-The Decoder
123
1
Burst Error Correction Circuitry-The Encoder
124
3
Using a Cyclic Hamming Code for Detecting Burst Errors
127
4
Using a Cyclic Hamming Code for Correcting Burst Errors Based on Knowing the Burst Pattern
131
2
The Fire Code
133
4
Single Character Error Correction (the RS Code)
137
10
Convolutional Codes
147
15
Basic Concepts
147
2
Diffuse Threshold Decoding Codes: Encoding Process
149
1
Diffuse Threshold Decoding Codes: Decoding Process
150
12
Appendix: The Basic Principles Re-Examined
162
43
A.1 Polynomial Division Modulo 2
162
2
A.2 The connection between Polynomial Division and LFSR Circuitry
164
10
A.3 The Generating Polynomial of a Code
174
5
A.4 Cyclic Properties of Polynomials
179
4
A.5 More Properties of Cyclic Codes Based on Polynomial Manipulations
183
9
A.6 A Formal Treartment of a Single Character Correction RS Code
192
13
Index
205