Stopping Redundancy Hierarchy Beyond the Minimum Distance
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Stopping sets play a crucial role in failure events of iterative decoders over a binary erasure channel (BEC). The l-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 ℓ. In this work, a notion of coverable stopping sets is defined. In order to achieve maximum-likelihood performance under iterative decoding over BEC, the parity-check matrix should contain no coverable stopping sets of size ℓ, for 1 <ℓ< 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 ℓ-th stopping redundancy, 1 <ℓ< n-k. The bounds are derived for both specific codes and codes ensembles. In the range 1 <ℓ< d-1, for specific codes, the new bounds improve on the results in the literature. Numerical results are also obtained.
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