On random-like codes

Abstract

A random-like code is designed so as to make its normalized distance distribution (its weight distribution if it is linear) close to that obtained in the average by random coding. The iterated product of single-parity-check codes, for testing the use of several successive weighted-output decodings in order to reduce the overall decoding complexity, was the first example of a random-like code with a poor minimum distance but a good bit-error rate.

A distinction is introduced between strongly and weakly random-like codes. Strongly random-like codes are such that each term of their weight distribution is close to that of random coding, in contrast with weakly random-like ones which exhibit only a global shape similarity (as measured e.g., by the cross-entropy). Strongly random-like codes are good for the minimum distance criterion as meeting the Gilbert-Varshamov bound and their word-error rate (WER) is good. In contrast, weakly random-like codes may have small bit-error rate (BER) but bad WER. From an engineering point of view, however, the BER is the most significant parameter so weakly random-like codes are actually interesting. Moreover, their design is easier.

We briefly review some codes which may be considered as random-like, with special emphasis on so-called “pseudo-random” systematic convolutional codes which are recursive and such that the recursive part of the encoder corresponds to a maximum-length generator. Even if such a code has a small free distance and thus does not result in a low enough BER, combining several such codes according to the turbo-code scheme enables control of the weight distribution tails, hence the BER, by increasing the number of combined codes, so as to meet any specified performance at a rate lower than the channel capacity.