Improved linear programming decoding and bounds on the minimum distance of LDPC codes

We propose a technique for improving LP decoding, based on the merging of check nodes. This technique can be applied to standard as well as generalized LDPC codes. Furthermore, we show how a recently-discovered linear-complexity LP decoder can be used to derive non-trivial lower bounds on the minimum distance of specific LDPC codes, with complexity that exhibits quadratic growth with respect to the block length. This bound can be refined using the check node merging technique. The lower bound on the minimum distance is shown to be an upper bound on the fractional distance of the code.