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The "Divide and Concur'' (DC) algorithm, recently introduced by Gravel andElser, can be considered a competitor to the belief propagation (BP) algorithm,in that both algorithms can be applied to a wide variety of constraintsatisfaction, optimization, and probabilistic inference problems. We show thatDC can be interpreted as a message-passing algorithm on a constraint graph,which helps make the comparison with BP more clear. The "difference-map''dynamics of the DC algorithm enables it to avoid "traps'' which may be relatedto the "trapping sets'' or "pseudo-codewords'' that plague BP decoders oflow-density parity check (LDPC) codes in the error-floor regime.  We investigate two decoders for low-density parity-check (LDPC) codes basedon these ideas. The first decoder is based directly on DC, while the seconddecoder borrows the important "difference-map'' concept from the DC algorithmand translates it into a BP-like decoder. We show that this "difference-mapbelief propagation'' (DMBP) decoder has dramatically improved error-floorperformance compared to standard BP decoders, while maintaining a similarcomputational complexity. We present simulation results for LDPC codes on theadditive white Gaussian noise and binary symmetric channels, comparing DC andDMBP decoders with other decoders based on BP, linear programming, andmixed-integer linear programming.

Divide & Concur and Difference-Map BP Decoders for LDPC Codes