LEE-METRIC BCH CODES AND THEIR APPLICATION TO CONSTRAINED AND PARTIAL-RESPONSE CHANNELS

We show that each code in a certain class of BCH codes over GF(p), spe cified by a code length n less-than-or-equal-to p(m) - 1 and a runleng th r less-than-or-equal-to (p - 1)/2 of consecutive roots in GF(p(m)), has minimum Lee distance greater-than-or-equal-to 2r. For the very hi gh-rate range these codes approach the sphere-packing bound on the min imum Lee distance. Furthermore, for a given r, the length range of the se 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 des igned minimum Lee distance 2r. Bounds on the minimum Lee distance for r greater-than-or-equal-to (p + 1)/2 are provided for the Reed-Solomon case, i.e., when the BCH code roots are in GF(p). We present two appl ications. First, Lee-metric BCH codes can be used for protecting again st bitshift errors and synchronization errors caused by insertion and / or deletion of zeros in (d, k)-constrained channels. Second, the cod e 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.