The method is very simple: it first applies ordinary LP decoding, and when it fails, it proceeds by guessing facets of the polytope, and then resolving the linear program on these facets. We further exploit these geometrical properties to devise an improved decoding algorithm with the same order of complexity as LP decoding that provably performs better. We then prove that for expander codes, the active set of any fractional pseudocodeword is smaller by a constant fraction than that of any codeword. We first show that for expander codes, every fractional pseudocodeword always has at least a constant fraction of nonintegral bits. We investigate the structure of the polytope underlying the linear programming (LP) decoder introduced by Feldman, Karger, and Wainwright.
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