Book description
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.
Table of contents
- Cover
- Title Page
- Copyright Page
- Dedication
- Table of Contents
- Preface
- Notations and Abbreviations
- 1 Historical Background
-
2 Digital Arithmetic
- 2.1 Number Systems
-
2.2 Boolean and Bitwise Operations
- 2.2.1 Boolean Logical Operations
- 2.2.2 Bitwise Operations
- 2.2.3 Applications
- 2.2.4 XOR Swap Algorithm
- 2.2.5 XOR Linked Lists
- 2.2.6 Bit Shifts
- 2.2.7 Arithmetic Shifts
- 2.2.8 Logical Shifts
- 2.2.9 Circular Shifts
- 2.2.10 Shift Registers
- 2.2.11 Rotate through Carry
- 2.2.12 One’s and Two’s Complement Arithmetic
- 2.3 Checksum
- 2.4 Ring Counters
- 2.5 Residues, Residue Classes, and Congruences
- 2.6 Integral Approximations
- 2.7 Lexicographic Order
-
3 Linear Codes
- 3.1 Linear Vector Spaces over Finite Fields
- 3.2 Communication Channels
- 3.4 Linear Codes
- 3.5 Vector Operations
- 3.6 Sphere Packing
- 4 Hamming Codes
- 5 Extended Hamming Codes
-
6 Bounds in Coding Theory
- 6.1 Definitions
- 6.2 Sphere-Packing Bound
- 6.3 Johnson Bound
- 6.4 Gilbert-Varshamov Bound
- 6.5 Hamming Bound
- 6.6 Singleton Bound
- 6.7 Plotkin Bound
- 6.8 Griesmer Bound
- 6.9 Zyablov Bound
- 6.10 Bounds in Fn2
- 6.11 Reiger Bound
- 6.12 Krawtchouk Polynomials
- 6.13 Linear Programming Bound
- 6.14 Stochastic Bounds for SEC-DED Codes
- 7 Golay Codes
- 8 Galois Fields
- 9 Matrix Codes
- 10 Cyclic Codes
- 11 BCH Codes
- 12 Reed-Muller Codes
- 13 Reed–Solomon Codes
- 14 Belief Propagation
- 15 LDPC Codes
- 16 Special LDPC Codes
- 17 Discrete Distributions
- 18 Erasure Codes
- 19 Luby Transform Codes
- 20 Raptor Codes
- A ASCII Table
- B Some Useful Groups
- C Tables in Finite Fields
- D Discrete Fourier Transform
- E Software Resources
- Bibliography
- Index
Product information
- Title: Algebraic and Stochastic Coding Theory
- Author(s):
- Release date: July 2017
- Publisher(s): CRC Press
- ISBN: 9781466505629
You might also like
book
Advanced Theory of Constraint and Motion Analysis for Robot Mechanisms
provides a complete analytical approach to the invention of new robot mechanisms and the analysis of …
book
Adaptive Learning Methods for Nonlinear System Modeling
Adaptive Learning Methods for Nonlinear System Modeling presents some of the recent advances on adaptive algorithms …
book
A First Course in Mathematical Logic and Set Theory
A mathematical introduction to the theory and applications of logic and set theory with an emphasis …
book
Optimization Techniques for Solving Complex Problems
Real-world problems and modern optimization techniques to solve them Here, a team of international experts brings …