Algebraic and Stochastic Coding Theory

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CRC Press, Mar 5, 2012 - Computers - 512 pages
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Using a simple yet rigorous approach, Algebraic and Stochastic Coding Theory makes the subject of coding theory easy to understand for readers with a thorough knowledge of digital arithmetic, Boolean and modern algebra, and probability theory. It explains the underlying principles of coding theory and offers a clear, detailed description of each code. More advanced readers will appreciate its coverage of recent developments in coding theory and stochastic processes.

After a brief review of coding history and Boolean algebra, the book introduces linear codes, including Hamming and Golay codes. It then examines codes based on the Galois field theory as well as their application in BCH and especially the Reed–Solomon codes that have been used for error correction of data transmissions in space missions.

The major outlook in coding theory seems to be geared toward stochastic processes, and this book takes a bold step in this direction. As research focuses on error correction and recovery of erasures, the book discusses belief propagation and distributions. It examines the low-density parity-check and erasure codes that have opened up new approaches to improve wide-area network data transmission. It also describes modern codes, such as the Luby transform and Raptor codes, that are enabling new directions in high-speed transmission of very large data to multiple users.

This robust, self-contained text fully explains coding problems, illustrating them with more than 200 examples. Combining theory and computational techniques, it will appeal not only to students but also to industry professionals, researchers, and academics in areas such as coding theory and signal and image processing.

 

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Contents

1 Historical Background
1
2 Digital Arithmetic
15
3 Linear Codes
43
4 Hamming Codes
75
5 Extended Hamming Codes
95
6 Bounds in Coding Theory
117
7 Golay Codes
143
8 Galois Fields
161
15 LDPC Codes
307
16 Special LDPC Codes
327
17 Discrete Distributions
355
18 Erasure Codes
375
19 Luby Transform Codes
395
20 Raptor Codes
411
A ASCII Table
427
B Some Useful Groups
431

9 Matrix Codes
181
10 Cyclic Codes
203
11 BCH Codes
217
12 Reed8211Muller Codes
231
13 Reed8211Solomon Codes
257
14 Belief Propagation
287
C Tables in Finite Fields
441
D Discrete Fourier Transform
447
E Software Resources
457
Bibliography
461
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