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IEEE International Symposium on Information Theory (ISIT)

On the asymptotic weight distribution of regular LDPC ensembles

Vishwambhar Rathi

Published Jan 1, 2005• 11 citations
Publisher

Abstract

We estimate the variance of weight 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 close to the ensemble average. We are able to show that a large fraction of total number of codes have their weight distribution close to the average

DOI
10.1109/isit.2005.1523729

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We estimate the variance of weight 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 close to the ensemble average. We are able to show that a large fraction of total number of codes have their weight distribution close to the average
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The average weight distribution of a regular low-density parity-check (LDPC) code ensemble over a finite field is thoroughly analyzed. In particular, a precise asymptotic approximation of the average weight distribution is derived for the small-weight case, and a series of fundamental qualitative properties of the asymptotic growth rate of the average weight distribution are proved. Based on this analysis, a general result, including all previous results as special cases, is established for the minimum distance of individual codes in a regular LDPC code ensemble.
We consider the ensemble of regular LDPC codes and obtain an expression for the second moment of the distance spectrum. We show how this expression can be used to derive a lower bound on the probability that the growth rate of a randomly chosen code from the ensemble is equal to the growth rate of the average distance spectrum, when the block length is sufficiently large. In particular, when the connectivity of the code is sufficiently large, the distance spectrum of a code in the ensemble is concentrated. We then derive a lower bound on the probability (confidence level) that the minimum distance and error rate, respectively, of a randomly chosen code from the ensemble are upper and lower bounded by some values (which depend on the confidence level)
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