  Recently there has been interest in the construction of small parity check
sets for iterative decoding of the Hamming code with the property that each
uncorrectable (or stopping) set of size three is the support of a codeword and
hence uncorrectable anyway. Here we reformulate and generalise the problem, and
improve on this construction. First we show that a parity check collection that
corrects all correctable erasure patterns of size m for the r-th order Hamming
code (i.e, the Hamming code with codimension r) provides for all codes of
codimension $r$ a corresponding ``generic'' parity check collection with this
property. This leads naturally to a necessary and sufficient condition on such
generic parity check collections. We use this condition to construct a generic
parity check collection for codes of codimension r correcting all correctable
erasure patterns of size at most m, for all r and m <= r, thus generalising the
known construction for m=3. Then we discussoptimality of our construction and
show that it can be improved for m>=3 and r large enough. Finally we discuss
some directions for further research.
