Decoding of Cyclic Codes over Symbol-Pair Read Channels

Abstract

Symbol-pair read channels, in which the outputs of the read process are pairs of consecutive symbols, were recently studied by Cassuto and Blaum. This new paradigm is motivated by the limitations of the reading process in high density data storage systems. They studied error correction in this new paradigm, specifically, the relationship between the minimum Hamming distance of an error correcting code and the minimum pair distance, which is the minimum Hamming distance between symbol-pair vectors derived from codewords of the code. It was proved that for a linear cyclic code with minimum Hamming distance d_H, the corresponding minimum pair distance is at least d_H + 3. Our main contribution is proving that, for a given linear cyclic code with a minimum Hamming distance d_H, the minimum pair distance is at least d_H + [dH/2]. We also describe decoding algorithms, based upon bounded distance decoders for the cyclic code, whose pair-symbol error correcting capabilities reflects the larger minimum pair distance. In addition, we consider the case where a read channel output is a prescribed number, b > 2, of consecutive symbols and provide some generalizations of our results. We note that the symbol-pair read channel problem is a special case of the sequence reconstruction problem that was introduced by Levenshtein.