ML decoding via mixed-integer adaptive linear programming

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Linear programming (LP) decoding was introduced by Feldman et al. (IEEE Trans. Inform. Theory Mar. 2005) as a novel way to decode binary low-density parity-check codes. Taghavi and Siegel (Proc. ISIT 2006) describe a computationally simplified decoding approach they term "adaptive" LP decoding. Adaptive LP decoding starts with a sub-set of the LP constraints, and iteratively adds violated constraints until an optimum of the original LP is found. Usually only a tiny fraction of the original constraints need to be reinstated, leading to huge efficiency gains compared to ordinary LP decoding. Here we describe a modification of the adaptive LP decoder that results in a maximum likelihood (ML) decoder. Whenever the adaptive LP decoder returns a pseudo-codeword rather than a codeword, we add an integer constraint on the least certain symbol of the pseudo-codeword. For certain codes, and especially in the high-SNR (error floor) regime, only a few integer constraints are required to force the resultant mixed-integer LP to the ML solution. We demonstrate that our approach can efficiently achieve the optimal ML decoding performance on a (155,64) LDPC code introduced by Tanner et al.