Stopping set analysis of repeat multiple-accumulate codes

Eirik Rosnes, Alexandre Graell i Amat

Abstract: In this work, we consider a stopping set analysis of repeat multiple-accumulate (RMA) code ensembles formed by the serial concatenation of a repetition code with multiple accumulators. The RMA codes are assumed to be iteratively decoded in a constituent code oriented fashion using maximum a posteriori erasure correction in the constituent codes. We give stopping set enumerators for RMA code ensembles and show that their stopping distance h <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</inf> , defined as the size of the smallest nonempty stopping set, asymptotically grows linearly with the block length. Thus, the RMA code ensembles are good for the binary erasure channel. Furthermore, it is shown that, contrary to the asymptotic minimum distance d <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</inf> , whose growth rate coefficient increases with the number of accumulate codes, the h <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</inf> growth rate coefficient diminishes with the number of accumulators. We also consider random puncturing and show that for sufficiently high code rates, the asymptotic h <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</inf> does not grow linearly with the block length, contrary to the asymptotic d <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</inf> , whose growth rate coefficient approaches the Gilbert-Varshamov bound as the rate increases. Finally, we give iterative decoding thresholds to show the convergence properties.