The serial concatenation of rate-1 codes through uniform random interleavers

Until the analysis of repeat accumulate codes by Divsalar et al. (1998), few people would have guessed that simple rate-1 codes could play a crucial role in the construction of "good" binary codes. We construct "good" binary linear block codes at any rate r<1 by serially concatenating an arbitrary outer code of rate r with a large number of rate-1 inner codes through uniform random interleavers. We derive the average output weight enumerator (WE) for this ensemble in the limit as the number of inner codes goes to infinity. Using a probabilistic upper bound on the minimum distance, we prove that long codes from this ensemble will achieve the Gilbert-Varshamov (1952) bound with high probability. Numerical evaluation of the minimum distance shows that the asymptotic bound can be achieved with a small number of inner codes. In essence, this construction produces codes with good distance properties which are also compatible with iterative "turbo" style decoding. For selected codes, we also present bounds on the probability of maximum-likelihood decoding (MLD) error and simulation results for the probability of iterative decoding error.