LP Decoding of Regular LDPC Codes in Memoryless Channels

Abstract

We study error bounds for linear programming decoding of regular low-density parity-check (LDPC) codes. For memoryless binary-input output-symmetric channels, we prove bounds on the word error probability that are inverse doubly exponential in the girth of the factor graph. For memoryless binary-input AWGN channel, we prove lower bounds on the threshold for regular LDPC codes whose factor graphs have logarithmic girth under LP-decoding. Specifically, we prove a lower bound of σ = 0.735 (upper bound of [(Eb)/(N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> )]=2.67 dB) on the threshold of (3, 6)-regular LDPC codes whose factor graphs have logarithmic girth. Our proof is an extension of a recent paper of Arora, Daskalakis, and Steurer [STOC 2009] who presented a novel probabilistic analysis of LP decoding over a binary symmetric channel. Their analysis is based on the primal LP representation and has an explicit connection to message passing algorithms. We extend this analysis to any MBIOS channel.