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.