Chapter Overview

Every engineered communication system, ranging from satellite communications to hard disk drives to Ethernet must operate under noisy conditions. The key to reliable communication and storage is to add an appropriate amount of redundancy to make the system reliable. The field of channel coding is concerned with constructing channel codes and their decoding algorithms: controlled redundancy is introduced into a message prior to its transmission over a noisy channel (the encoding step), and this redundancy is removed from the received noisy string to unveil the intended message (the decoding step). The encoded message is referred to as the codeword. The collection of all codewords is a channel code. Assuming all the messages have the same length, and all the codewords have the same length, the ratio of message length to codeword length is the code rate. To make coding systems implementable in practice, channel codes must provide the best possible protection to noise while their decoding algorithms must be of acceptable complexity.

There is a clear tension with this dual goal: if a channel code protects a fairly long encoded message with relatively few but carefully derived redundancy bits (necessary for high performance), the optimal, maximum likelihood decoding algorithm has exponential complexity.