Iterative Receiver Design
Exploiting the power of factor graphs, this detailed text provides a general framework for systematically developing iterative algorithms for digital receivers, and highlights connections between important algorithms.
TABLE OF CONTENTS Iterative Receiver Design
8.3: Block Codes
8.3 Block Codes
Before we delve into specific types of error-correcting codes, let us first review some basic terminology, show how codes can be related to factor graphs, and introduce the concept of puncturing.
8.3.1 Basic concepts
We will limit ourselves to binary codes, where the components of b and c belong to 
. The binary field contains two elements, commonly denoted 0 and 1. The field is endowed with two operators: addition (+) and multiplication ( ), defined in Table 8.1. Multiplication is distributive over addition.
| b 1 | b 2 | b 1 + b 2 | b 1 b 2 |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 0 | 1 |
A code is defined as a set of codewords 
, and the field of coding theory is mainly concerned with searching and analyzing these sets.For practical purposes, we also need a function that maps an information word b onto a codeword c ? 
: c = f c ( b). This is a process known as encoding. The encoding process is reversible, so, for every codeword c ? , we can write 
for some 
.
Definition 8.1: (Block code).
An ( N c, N b) block code is a set of 
distinct elements, with N
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