Some Z(n+2) terraces from Z(n) power-sequences, n being an odd prime

A terrace for Z(m) is an arrangement (a(1), a(2), . . . , a(m)) of the m elements of Z(m) such that the sets of differences a(i +) (1) - a(i) and a(i) - a(i +) (1) (i = 1, 2, . . . , m - 1) between them contain each element of Z(m)\{0} exactly twice. For m odd, many procedures are available for constructing power-sequence terraces for Z(m); each terrace of this sort may be partitioned into segments one of which contains merely the zero element of Z(m), whereas each other segment is either (a) a sequence of successive powers of an element of Z(m) or (b) such a sequence multiplied throughout by a constant. We now extend this idea by using power-sequences in Z(n), where n is an odd prime, to obtain terraces for Z(m) where m = n + 2. Our technique needs each of the n - 1 elements from Z(n)\{0} to be written so as to lie in the interval (0, n) and for three further elements 0, n and n + I then to be introduced. A segment of one of the new terraces may contain just a single element from the set g(0) = {0, n, n +1} or it may be of type (a) or (b) with m = n and containing successive powers of 2, each evaluated modulo n. Also, a segment based on successive powers of 2 may be broken in one, two or three places by putting a different element from g(0) in each break. We provide Z(n+2) terraces for all odd primes n satisfying 0 < n < 1000 except for n = 127, 601, 683.