PERFECT (D, K)-CODES CAPABLE OF CORRECTING SINGLE PEAK-SHIFTS

Codes, consisting of sequences 0 (alpha1) 10 (alpha2)1 ... 0 (alpha N)1, where d less-than-or-equal-to alpha(i) less-than-or-equal-to k, and call them (d,k)-codes of reduced length N are considered. We introduce a definition of arbitrary (d, k)- and perfect (d, k)-codes capable of correcting single peak-shifts of given size t. 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 not only effective, but also universal in the sense that it does not depend on the (d, k)-constraints. It also allows to correct automatically those peak-shifts that violate (d, k)-constraints. Furthermore, our construction is naturally extended to (d, k)-codes of fixed binary length and allows the determination of the beginning of the next codeword. The question whether the designed codes can be represented as systematic codes with minimal redundancy is considered as well. In particular, for any (d, k)-code with n q-ary (q = k - d + 1 > 2) information digits, a method of finding r q-ary check digits is given such that the resulting systematic code is capable of correcting single peak-shifts of size 1, where r is determined uniquely by q(r - 1) - 2(r - 1) < 2n + 1 less-than-or-equal-to q(r) - 2r. This code is perfect if 2n + 1 = q(r) - 2r.