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In this paper, redundant random ensembles are defined and their average 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 significantly reduce the number of stopping sets (SS) of small size. Upper and lower bounds on the average SS weight distribution of the redundant random ensembles are proved based on a combinatorial argument. Asymptotic forms of these bounds reveal asymptotic behavior of the average SS weight distributions. From these bounds, a tradeoff between the number of redundant rows (corresponding to decoding complexity of belief propagation on binary erasure channel) and the average SS weight distribution (corresponding to decoding performance) can be derived.