Two-dimensional burst identification codes and their use in burst correction

A new class of codes, called burst identification codes, is defined and studied. These codes can be used to determine the patterns of burst errors. Two-dimensional burst correcting codes can be easily constructed from burst identification codes. The resulting class of codes is simple to implement and has lower redundancy than other comparable codes. The results are pertinent to the study of radiation effects on VLSI RAM chips, which can cause two-dimensional bursts of errors

On the existence of optimum cyclic burst-correcting codes

It is shown that for each integer b >= 1 infinitely many optimum cyclic b-burst-correcting codes exist, i.e., codes whose length n, redundancy r, and burst-correcting capability b, satisfy n = 2^{r-b+1} - 1. Some optimum codes for b = 3, 4, and 5 are also studied in detail

Codes for correcting three or more adjacent deletions or insertions

Codes are presented that can correct the deletion or the insertion of a predetermined number of adjacent bits greater than or equal to three. This extends the constructions of codes beyond those proposed by Levenshtein fifty years ago to correct one or two adjacent deletions or insertions

Moment balancing templates: Universal constructions to add insertion/deletion correction capability to arbitrary error correcting or constrained codes

Abstract: We investigate extending a chosen block or convolutional code which has additive error correction capability, as predetermined by the usual communication systems or coding considerations. Our extension involves constructing a template to add additional redundant bits in positions, selected to balance the moment of the code word. Using some number theoretic constructions in the literature, insertion/deletion correction can then be achieved. If the template is carefully designed, the number of additional redundant bits for the insertion/deletion correction can be kept small - in some cases of the same order as for Hamming codes. Our construction technique can also be used for the systematic encoding of number theoretic codes, and furthermore have implications for other coding techniques utilizing the moment function, such as codes correcting asymmetrical errors, spectral shaping codes, or constant weight codes

Moment balancing templates: constructions to add insertion/deletion correction capability to error correcting or constrained codes

Abstract: Templates are constructed to extend arbitrary additive error correcting or constrained codes, i.e., additional redundant bits are added in selected positions to balance the moment of the codeword. The original codes may have error correcting capabilities or constrained output symbols as predetermined by the usual communication system considerations, which are retained after extending the code. Using some number theoretic constructions in the literature, insertion/deletion correction can then be achieved. If the template is carefully designed, the number of additional redundant bits for the insertion/deletion correction can be kept small—in some cases of the same order as the number of parity bits in a Hamming code of comparable length

Generalized stopping sets and stopping redundancy

Abstract — Iterative decoding for linear block codes over erasure channels may be much simpler than optimal decoding but its performance is usually not as good. Here, we present a general iterative decoding technique that gives a more refined tradeoff between complexity and performance. In each iteration, a system of equations is solved. In case the maximum number of equations to be solved is just one, the general iterative decoder reduces to the well-known iterative decoder. On the other hand, if the maximum number is set to the redundancy of the codes, the general iterative decoder gives the same performance as the optimal decoder. Varying the maximum number of equations to be solved in each iteration between these two extremes allows for a better match, in terms of performance and complexity, to the system specifications. Stopping sets and stopping redundancy are important concepts in the analysis of the performance and complexity of iterative decoders on the erasure channel. In consequence of the new generalized decoding procedure, the notions of stopping sets and stopping redundancy are generalized as well. Basic properties and examples of both generalized stopping sets and generalized stopping redundancy are presented in this paper. I