Stopping Redundancy Hierarchy Beyond the Minimum Distance

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.