Good concatenated code ensembles for the binary erasure channel

In this work, we give good concatenated code ensembles for the binary erasure channel (BEC). In particular, we consider repeat multiple-accumulate (RMA) code ensembles formed by the serial concatenation of a repetition code with multiple accumulators, and the hybrid concatenated code (HCC) ensembles recently introduced by Koller et al. (5th Int. Symp. on Turbo Codes & Rel. Topics, Lausanne, Switzerland) consisting of an outer multiple parallel concatenated code serially concatenated with an inner accumulator. We introduce stopping sets for iterative constituent code oriented decoding using maximum a posteriori erasure correction in the constituent codes. We then analyze the asymptotic stopping set distribution for RMA and HCC ensembles and show that their stopping distance h_min, defined as the size of the smallest nonempty stopping set, asymptotically grows linearly with the block length. Thus, these code ensembles are good for the BEC. It is shown that for RMA code ensembles, contrary to the asymptotic minimum distance d_min, whose growth rate coefficient increases with the number of accumulate codes, the h_min growth rate coefficient diminishes with the number of accumulators. We also consider random puncturing of RMA code ensembles and show that for sufficiently high code rates, the asymptotic h_min does not grow linearly with the block length, contrary to the asymptotic d_min, whose growth rate coefficient approaches the Gilbert-Varshamov bound as the rate increases. Finally, we give iterative decoding thresholds for the different code ensembles to compare the convergence properties.

I. Introduction

Turbo codes introduced by Berrou et al. in [1], and Gallager's low-density parity-check (LDPC) codes [2], rediscovered by MacKay in [3], are considered among the most powerful error-correction schemes of today due to their low decoding complexity and near-capacity performance on a wide variety of channels. Recently, serially concatenated codes, such as very simple repeat accumulate (RA) codes, and hybrid concatenated codes (HCCs), i.e., mixed parallel and serial structures combining the features of the two concatenations [4]–[13], have also attracted some attention, since they yield better minimum distances than turbo codes.