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

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.Comment: 13 pages, no figures. Submitted to IEEE Transactions on Information Theory, July 28, 200

On q-ary codes correcting all unidirectional errors of a limited magnitude

We consider codes over the alphabet Q={0,1,..,q-1}intended for the control of unidirectional errors of level l. That is, the transmission channel is such that the received word cannot contain both a component larger than the transmitted one and a component smaller than the transmitted one. Moreover, the absolute value of the difference between a transmitted component and its received version is at most l. We introduce and study q-ary codes capable of correcting all unidirectional errors of level l. Lower and upper bounds for the maximal size of those codes are presented. We also study codes for this aim that are defined by a single equation on the codeword coordinates(similar to the Varshamov-Tenengolts codes for correcting binary asymmetric errors). We finally consider the problem of detecting all unidirectional errors of level l.Comment: 22 pages,no figures. Accepted for publication of Journal of Armenian Academy of Sciences, special issue dedicated to Rom Varshamo

An assessment of envelope-based demodulation in case of proximity of carrier and modulation frequencies

If C is a q-ary code of length n and a and b are two codewords, then c is called a descendant of a and b if c i 2 fa i ; b i g for i = 1; : : : ; n. We are interested in codes C with the property that, given any descendant c, one can always identify at least one of the `parent&apos; codewords in C. We study bounds on F (n; q), the maximal cardinality of a code C with this property, which we call the identifiable parent property. Such codes play a role in schemes that protect against piracy of software