Lee-Metric BCH Codes and their Application to Constrained and Partial-Response Channels

Abstract

We show that each code in a certain class of BCH codes over GF(p), specified by a code length n ≤ pm - 1 and a runlength r ≤ (p - l)/2 of consecutive roots in GF(pm), 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 extended negacyclic codes. We 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). We 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. © 1994 IEEE