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The “Divide and Concur” (DC) algorithm introduced by Gravel and Elser can be considered a competitor to the belief propagation (BP) algorithm, in that both algorithms can be applied to a wide variety of constraint satisfaction, optimization, and inference problems. We show that DC can be interpreted as a message-passing algorithm on a “normal” factor graph. The “difference-map” dynamics of the DC algorithm enables it to avoid “traps” which may be related to the “trapping sets” or “pseudo-codewords” that plague BP decoders of low-density parity check (LDPC) codes in the error-floor regime. We investigate two decoders for LDPC codes based on these ideas. The first decoder is based directly on DC, while the second decoder borrows the important “difference-map” concept from the DC algorithm and translates it into a BP-like decoder. We show that this “difference-map belief propagation” (DMBP) decoder has dramatically improved error-floor performance compared to standard BP decoders, while maintaining a similar computational complexity. We present simulation results for LDPC codes comparing DC and DMBP decoders with other decoders based on sum-product BP, linear programming, and mixed-integer linear programming. We also describe the close relation of the DMBP decoder to reweighted min-sum algorithms, including those recently proposed by Ruozzi and Tatikonda.