A conceptual framework for understanding turbo codes

For understanding turbo codes, we propose to locate them at the intersection of three main topics: the random-like criterion for designing codes, the idea of extending the decoding role to reassess probabilities, and that of combining several codes by product or concatenation. Concerning the idea of designing random-like (RL) codes, we distinguish strongly and weakly random-like codes depending on how the closeness of their weight distribution to that obtained in the average by random coding is measured. Using, e.g., the cross entropy as a closeness measure results in weakly RL codes. Although their word-error rate is bad, their bit-error rate (BER) remains low up to the vicinity of the channel capacity. We show that pseudorandom recursive convolutional codes belong to this family. Obtaining reasonably good performance with a single code of this type involves high complexity, and its specific decoding is difficult. However, using these codes as components in the turbo-code scheme is a simple means for improving the low-weight tail of the distribution and to adjust the BER to any specification. In order to increase the encoder memory without inordinate complexity, it is suggested to use iterated nonexhaustive replication decoding