Prerequisites

Before You Begin

This chapter builds on linear algebra (Chapter 1), probability and random processes (Chapter 2), digital modulation (Chapter 8), detection and estimation theory (Chapter 9), and information theory (Chapter 11). Generator and parity-check matrices require matrix algebra from Chapter 1. Performance analysis uses the Q-function and error probability framework from Chapter 9. Coding bounds and capacity-approaching arguments rely on the channel capacity concepts from Chapter 11.

Matrix operations over finite fields (binary arithmetic)(Review ch01)
Self-check: Can you perform matrix-vector multiplication over $\mathrm{GF}(2)$ (i.e., with modulo-2 addition and multiplication)?
Probability, Bayes rule, and random variables(Review ch02)
Self-check: Can you compute posterior probabilities using Bayes' rule and work with log-likelihood ratios?
Digital modulation and BER in AWGN(Review ch08)
Self-check: Can you compute the uncoded BER for BPSK as $P_b = Q(\sqrt{2E_b/N_0})$ ?
ML and MAP detection(Review ch09)
Self-check: Can you derive the ML decision rule for binary hypothesis testing and compute the resulting error probability?
Channel capacity and the Shannon limit(Review ch11)
Self-check: Can you state the channel capacity of the binary-input AWGN channel and explain what it means for a code to be "capacity-approaching"?

Chapter 12 Notation

Key symbols introduced or heavily used in this chapter.

Symbol	Meaning	Introduced
$n$	Block code length (number of coded bits)	s01
$k$	Number of information bits per block	s01
$R_c = k/n$	Code rate	s01
$d_{\min}$	Minimum Hamming distance of a code	s01
$t$	Error-correcting capability, $t = \lfloor (d_{\min}-1)/2 \rfloor$	s01
$\mathbf{G}$	Generator matrix ( $k \times n$ )	s01
$\mathbf{H}$	Parity-check matrix ( $(n-k) \times n$ )	s01
$K$	Constraint length of a convolutional code	s02
$d_{\text{free}}$	Free distance of a convolutional code	s02
$L(u)$	Log-likelihood ratio for bit $u$	s03
$I_A, I_E$	A priori and extrinsic mutual information (EXIT chart)	s03
$\gamma_c$	Coding gain (dB)	s01
$C_{\text{BICM}}$	BICM capacity (bits/channel use)	s06
$N_{\text{iter}}$	Number of decoding iterations	s03
$D$	Interleaving depth	s08

← Ch 11 Fundamentals of Error-Correcting Codes