Lee-metric BCH codes and their application to constrained and partial-response channels

Shows that each code in a certain class of BCH codes over GF(p), specified by a code length n⩽p^m-1 and a runlength r⩽(p-1)/2 of consecutive roots in GF(p^m), has minimum Lee distance ⩾2r. For the very high-rate range these codes approach the sphere-packing bound on the minimum Lee distance. Furthermore, for a given r, the length range of these codes is twice as large as that attainable by Berlekamp´s (1984) extended negacyclic codes. The authors present an efficient decoding procedure, based on Euclid´s algorithm, for correcting up to r-1 errors and detecting r errors, that is, up to the number of Lee errors guaranteed by the designed minimum Lee distance 2r. Bounds on the minimum Lee distance for r⩾(p+1)/2 are provided for the Reed-Solomon case, i.e., when the BCH code roots are in GF(p). The authors present two applications. First, Lee-metric BCH codes can be used for protecting against bitshift errors and synchronization errors caused by insertion and/or deletion of zeros in (d, k)-constrained channels. Second, the code construction with its decoding algorithm can be formulated over the integer ring, providing an algebraic approach to correcting errors in partial-response channels where matched spectral-null codes are used