Average Stopping Set Weight Distribution of Redundant Random Matrix Ensembles

TL;DR

This paper analyzes the stopping set weight distribution of redundant random matrix ensembles, revealing how redundancy affects decoding performance and complexity, and comparing their effectiveness to regular LDPC matrices.

Contribution

It introduces the concept of redundant random ensembles, derives bounds on their stopping set distributions, and explores the trade-offs between redundancy, decoding complexity, and performance.

Findings

Redundant random ensembles reduce small stopping sets.

Bounds on average SS weight distribution are established.

Dense matrices with redundancy can outperform regular LDPC matrices.

Abstract

In this paper, redundant random matrix ensembles (abbreviated as redundant random ensembles) are defined and their stopping set (SS) weight distributions are analyzed. A redundant random ensemble consists of a set of binary matrices with linearly dependent rows. These linearly dependent rows (redundant rows) significantly reduce the number of stopping sets of small size. An upper and lower bound on the average SS weight distribution of the redundant random ensembles are shown. From these bounds, the trade-off between the number of redundant rows (corresponding to decoding complexity of BP on BEC) and the critical exponent of the asymptotic growth rate of SS weight distribution (corresponding to decoding performance) can be derived. It is shown that, in some cases, a dense matrix with linearly dependent rows yields asymptotically (i.e., in the regime of small erasure probability) better performance than regular LDPC matrices with comparable parameters.