LP Decoding of Regular LDPC Codes in Memoryless Channels

TL;DR

This paper establishes error bounds and threshold estimates for linear programming decoding of regular LDPC codes over memoryless channels, extending previous analyses to a broader class of channels and providing concrete performance bounds.

Contribution

It extends probabilistic analysis of LP decoding to general MBIOS channels and derives bounds on error probability and thresholds for regular LDPC codes with logarithmic girth.

Findings

Inverse doubly-exponential error bounds in girth for certain channels

Lower bounds on decoding thresholds for regular LDPC codes

Extension of analysis to general MBIOS channels

Abstract

We study error bounds for linear programming decoding of regular 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 $\sigma=0.735$ (upper bound of $\frac{Eb}{N_0}=2.67$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.