Forward error correction via channel coding

Preface
Contents
Chapter 1: Review of Linear Algebra
	1.1 Group
		1.1.1 Subgroup
		1.1.2 Cosets
	1.2 Fields
		1.2.1 Binary or Galois Field
		1.2.2 Prime Fields
	1.3 Vector Spaces
		1.3.1 Vector Spaces
		1.3.2 Subspace
		1.3.3 Dual Space
		1.3.4 Linear Combinations
		1.3.5 Basis Vectors
		1.3.6 Matrices Obtained from Basis Vectors
		1.3.7 Polynomial Groups and Fields
	1.4 Ring
	Problems
Chapter 2: Linear Block Codes
	2.1 Binary Linear Block Codes
	2.2 Generator Matrix of a Linear Code
	2.3 Hamming Weight of a Code-Word
		2.3.1 Hamming Distance
		2.3.2 Minimum Distance of a Linear Block Code
	2.4 Performance Enhancement of Communication Systems and Encoding Operation
	2.5 The Philosophy of Channel Encoding
	2.6 Encoding and Decoding Operations
		2.6.1 Encoding Operation Using the Generator Matrix
	2.7 Dual Code
	2.8 Parity Check Matrix
	2.9 Systematic Form of a Generator Matrix
		2.9.1 Construction of Parity Check Matrix from Systematic Generator Matrix
	2.10 Equal and Equivalent Codes
	2.11 Finding the Minimum Distance of a Linear Block Code Using Its Parity Check Matrix
	2.12 Error Detection and Correction Capability of a Linear Block Code
	2.13 Hamming Spheres
		2.13.1 The Total Number of Words Inside a Hamming Sphere with Radius r
	2.14 Some Simple Bounds for the Minimum Distances of Linear Block Codes
	Problems
Chapter 3: Syndrome Decoding and Some Important Linear Block Codes
	3.1 Syndrome Decoding of Linear Block Codes
	3.2 Standard Array
	3.3 Error Correction Using Standard Array
	3.4 Syndrome
		3.4.1 Syndrome Table
	3.5 Syndrome Decoding
	3.6 Some Well-Known Linear Block Codes
		3.6.1 Single Parity Check Codes
		3.6.2 Repetition Codes
		3.6.3 Golay Code
		3.6.4 Reed-Muller Codes
		3.6.5 Hamming Codes
	3.7 Non-Systematic Form of Generator and Parity Check Matrices of the Hamming Codes
	3.8 Extended Hamming Codes
	3.9 Syndrome Decoding of Hamming Codes
	3.10 Shortened and Extended Linear Codes
	Problems
Chapter 4: Cyclic Codes
	4.1 Cyclic Codes
	4.2 Polynomials and Cyclic Codes
	4.3 The Generator Polynomial for Cyclic Codes
	4.4 Non-Systematic Encoding of Cyclic Codes
	4.5 Systematic Encoding of Cyclic Codes
		4.5.1 Code-Word in Systematic Form
	4.6 Decoding of Cyclic Codes
	4.7 Selection of Generator Polynomials of the Cyclic Codes
	4.8 Parity Check Polynomials of the Cyclic Codes
	4.9 Dual Cyclic Codes
	4.10 Generator and Parity Check Matrices for Cyclic Codes
	Problems
Chapter 5: Galois Fields
	5.1 Equation Roots and Concept of Field Extension
		5.1.1 Extension of Finite Fields
		5.1.2 Irreducible Polynomial
		5.1.3 Primitive Polynomial
	5.2 Construction of Extended Finite Fields
	5.3 Conjugate Classes
	5.4 Order of a Finite Field Element
	5.5 Minimal Polynomials
	5.6 Polynomials in Extended Fields
	5.7 Binary Representation of Extended Field Elements
	5.8 Equations in Extended Fields
	5.9 Matrices in Extended Fields
	Problems
Chapter 6: BCH Codes
	6.1 BCH Codes and Generator Polynomials of BCH Codes
	6.2 Parity Check and Generator Matrices of BCH Codes
		6.2.1 Second Method to Obtain the Generator and Parity Check Matrices of BCH Codes
	6.3 Syndrome Calculation for BCH Codes
	6.4 Syndrome Equations and Syndrome Decoding
		6.4.1 The Error Location Polynomial
		6.4.2 The Peterson-Gorenstein-Zierler (PGZ) Decoder
	Problems
Chapter 7: Reed-Solomon Codes
	7.1 Reed-Solomon Codes and Generator Polynomials of Reed-Solomon Codes
	7.2 Decoding of Reed-Solomon Codes
		7.2.1 Syndrome Decoding of Reed-Solomon Codes
		7.2.2 The Error Evaluator Polynomial
		7.2.3 Berlekamp Algorithm
	7.3 Generator Matrices of Reed-Solomon Codes
		7.3.1 Encoding of Reed-Solomon Codes
		7.3.2 Systematic Encoding Using Generator Polynomial
	7.4 Reed-Solomon Code as a Cyclic Code
		7.4.1 Parity Check Matrix of Reed-Solomon Code and Syndrome Calculation
		7.4.2 Parity Check Polynomials for RS Codes
		7.4.3 Chien Search
	Problems
Chapter 8: Convolutional Codes
	8.1 Convolutional Coding
		8.1.1 Convolutional Encoder Circuit
	8.2 Impulse Response of a Convolutional Encoder
		8.2.1 Short Method to Find the Impulse Response of a Convolutional Encoder
		8.2.2 Parallel Path Case
		8.2.3 Transfer Function Approach to Find the Impulse Responses of Convolutional Codes
		8.2.4 Convolution Operation
		8.2.5 Recursive Convolutional Encoders
	8.3 Generator Matrices for Convolutional Codes
	8.4 Polynomial Representation of Convolutional Codes
		8.4.1 Generator Matrix in Polynomial Form
	8.5 Generator Matrices of the Recursive Convolutional Encoders
	8.6 Graphical Representation of Convolutional Codes
		8.6.1 State Diagram Representation
		8.6.2 Trellis Diagram Representation of Convolutional Codes
	8.7 Decoding of Convolutional Codes
		8.7.1 Viterbi Algorithm
	Problems
	References
Index