LDPC code ensembles that universally achieve capacity under BP decoding: A simple derivation

A long-standing question in coding theory is whether code ensembles having a low-density parity check (LDPC) matrix can attain capacity under belief propagation (BP) decoding. An affirmative answer to this problem was recently given by the special class of spatially-coupled LDPC code ensemble. In this work, we provide a simple derivation of a different LDPC code ensemble that approaches capacity under BP decoding, following the classical approach of serial concatenation. This LDPC code ensemble is constructed by concatenating a high-rate outer LDPC code with an inner random convolutional one. The analysis of the concatenated-coding framework takes a particularly simple - “black box” - form. Specifically, the joint effect of the particular inner code and the binary-input memoryless output-symmetric (BMS) channel is encapsulated in a single parameter - the Bhattacharyya parameter, which is maximal for the binary symmetric channel (BSC). This implies that an inner convolutional code designed for the BSC achieves good performance over all BMS channels with a given capacity. Moreover, the performance guarantee of the outer LDPC code under BP decoding is dictated solely by this parameter. This, in turn, implies that the overall concatenated code approaches capacity under BP decoding for all BMS channels with a given capacity, simultaneously.