Multi-stage decoding of LDPC codes

In this paper we present a three-stage decoding strategy that combines quantized and un-quantized belief propagation (BP) decoders with a mixed-integer linear programming (MILP) decoder. Each decoding stage is activated only when the preceding stage fails to converge to a valid codeword. The faster BP decoding stages are able to correct most errors, yielding a short average decoding time. Only in the rare cases when the iterative stages fail is the slower but more powerfulMILP decoder used. The MILP decoder iteratively adds binary constraints until either the maximum likelihood codeword is found or some maximum number of binary constraints has been added. Simulation results demonstrate a large improvement in the word error rate (WER) of the proposed multi-stage decoder in comparison to belief propagation. The improvement is particularly noticeable in the low crossover probability (error floor) regime. Through introduction of an accelerated ¿active-set¿ version of the quantized BP decoder we significantly speed up the pace of simulation to simulate low density parity check (LDPC) codes of length up to around 2000 down to a WER of around 10^-10 on the binary symmetric channel. We demonstrate that for certain codes our approach can efficiently approach the optimal ML decoding performance for low crossover probabilities.