Perfect (d,k)-codes capable of correcting single peak-shifts

Abstract

Codes for the multibit peak-shift recording channel, called (d,k)-codes of reduced length N, are considered. Arbitrary (d,k)- and perfect (d,k)-codes capable of correcting single peak-shifts of given size t are defined. For the construction of perfect codes, a general combinatorial method connected with finding 'good' weight sequences in Abelian groups is used, and the concept of perfect t-shift N-designs is introduced. Explicit constructions of such designs for t=1, t=2, and t=(p-1)/2 are given, where p is a prime. This construction is universal in that it does not depend on the (d,k)-constraints. It also allows automatic correction of those peak-shifts that violate (d,k)-constraints. The construction is extended to (d,k)-codes of fixed binary length and allows the beginning of the next codeword to be determined. The question whether the designed codes can be represented as systematic codes with minimal redundancy is considered as well.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>