On parity check collections for iterative erasure decoding that correct all correctable erasure patterns of a given size

TL;DR

This paper develops a generalized method for constructing small parity check collections that can correct all correctable erasure patterns up to a certain size in Hamming codes, improving previous constructions and exploring optimality.

Contribution

It generalizes existing constructions for parity check collections to correct erasure patterns of size m in Hamming codes and provides a necessary and sufficient condition for their existence.

Findings

Constructed a generic parity check collection for all codes of a given codimension.

Established a necessary and sufficient condition for such collections.

Showed potential for improvement in the construction for larger r and m.

Abstract

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.