Generic erasure correcting sets: bounds and constructions
Journal of Combinatorial Theory Series A - Special issue in honor of Jacobus H. van Lint
Good concatenated code ensembles for the binary erasure channel
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TAMC'12 Proceedings of the 9th Annual international conference on Theory and Applications of Models of Computation
| Hi-index | 754.96 |
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 generalize the problem and improve on this construction. We show that a parity-check collection that corrects all correctable erasure patterns of size m for the Hamming code with codimension r provides, in fact, for all codes of codimension r a corresponding "generic" parity-check collection with this property. This leads in a natural way to a necessary and sufficient condition for 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 mlesr, thus generalizing the known construction for m=3. Then we discuss optimality of our construction and show that it can be improved for mges3 and r large enough. Finally, we discuss some directions for further research