Error Correcting and Detecting Codes

Abstract

As we have seen in Section 2.5, information theory provides theoretical upper bounds on the information rates that can be obtained over physical channels. While these bounds can be computed for a wide range of channels, the theory gives little indication of how they may be attained. It is fortuitous that, about the same time information theory was conceived, a theory of error correcting and detecting codes emerged. By the systematic injection of redundant bits into the encoding of information, the reliability of transmission could be improved. Hopes that codes achieving the bounds could be easily obtained proved to be illusionary; however, the theory has steadily developed so that codes have become indispensable components of many communications systems yielding considerable performance gains. The remarkable series of pictures that have been received from deep-space probes attest to the power of coding. A recent breakthrough, trellis coding, which is a direct outgrowth of the theory and which will be covered in Section 5.8, has put the Shannon bound within reach for bandlimited channels.

In writing this chapter significant use was made of the reference texts [1]–[4], particularly the first of these. In addition to these, we benefited from insightful review articles [5]–[8]. In keeping with the focus of the text, we shall consider codes in a communication context; however, a significant application of the technique lies in data storage and recording. (See [9] as an example ofthelauer.)