On the Asymptotic Weight and Stopping Set Distribution of Regular LDPC Ensembles

TL;DR

This paper analyzes the asymptotic behavior of weight and stopping set distributions in regular LDPC ensembles, providing bounds and showing most codes are close to ensemble averages.

Contribution

It introduces variance estimates and applies the second moment method to demonstrate most regular LDPC codes have typical weight and stopping set distributions.

Findings

Large fraction of codes have distributions close to ensemble averages

Bounds on probability of deviation from average distributions

Variance estimates enable probabilistic bounds

Abstract

We estimate the variance of weight and stopping set distribution of regular LDPC ensembles. Using this estimate and the second moment method we obtain bounds on the probability that a randomly chosen code from regular LDPC ensemble has its weight distribution and stopping set distribution close to respective ensemble averages. We are able to show that a large fraction of total number of codes have their weight and stopping set distribution close to the average.