  We investigate the stopping redundancy hierarchy of linear block codes and
its connection to permutation decoding techniques. An element in the ordered
list of stopping redundancy values represents the smallest number of possibly
linearly dependent rows in any parity-check matrix of a code that avoids
stopping sets of a given size. Redundant parity-check equations can be shown to
have a similar effect on decoding performance as permuting the coordinates of
the received codeword according to a selected set of automorphisms of the code.
Based on this finding we develop new decoding strategies for data transmission
over the binary erasure channel that combine iterative message passing and
permutation decoding in order to avoid errors confined to stopping sets. We
also introduce the notion of s-SAD sets, containing the smallest number of
automorphisms of a code with the property that they move any set of not more
than s erasures into positions that do not correspond to stopping sets within a
judiciously chosen parity-check matrix.
