Stopping Redundancy Hierarchy Beyond the Minimum Distance

TL;DR

This paper introduces the concept of coverable stopping sets and derives upper bounds on stopping redundancy, enhancing understanding of iterative decoding performance over the BEC for specific codes and ensembles.

Contribution

It defines coverable stopping sets and provides improved upper bounds on stopping redundancy, advancing the analysis of iterative decoding failure probabilities.

Findings

Upper bounds on stopping redundancy are derived for various codes.

New bounds improve upon existing results for certain code parameters.

Numerical calculations validate the theoretical bounds.

Abstract

Stopping sets play a crucial role in failure events of iterative decoders over a binary erasure channel (BEC). The $\ell$-th stopping redundancy is the minimum number of rows in the parity-check matrix of a code, which contains no stopping sets of size up to $\ell$. In this work, a notion of coverable stopping sets is defined. In order to achieve maximum-likelihood performance under iterative decoding over the BEC, the parity-check matrix should contain no coverable stopping sets of size $\ell$, for $1 \le \ell \le n-k$, where $n$ is the code length, $k$ is the code dimension. By estimating the number of coverable stopping sets, we obtain upper bounds on the $\ell$-th stopping redundancy, $1 \le \ell \le n-k$. The bounds are derived for both specific codes and code ensembles. In the range $1 \le \ell \le d-1$, for specific codes, the new bounds improve on the results in the literature. Numerical calculations are also presented.