A general approach to constructing power-sequence terraces for ZnZn

A terrace for ZnZn is an arrangement (a1,a2,…,an)(a1,a2,…,an) of the nn elements of ZnZn such that the sets of differences ai+1-aiai+1-ai and ai-ai+1ai-ai+1(i=1,2,…,n-1)(i=1,2,…,n-1) between them contain each element of Zn⧹{0}Zn⧹{0} exactly twice. For nn odd, many procedures have been published for constructing power-sequence terraces for ZnZn; each such terrace may be partitioned into segments one of which contains merely the zero element of ZnZn whereas each other segment is either (a) a sequence of successive powers of an element of ZnZn or (b) such a sequence multiplied throughout by a constant. We now present a new general power-sequence approach that yields ZnZn terraces for all odd primes nn less than 10001000 except for n=601n=601. It also yields terraces for some groups ZnZn with n=p2n=p2 where pp is an odd prime, and for some ZnZn with n=pqn=pq where pp and qq are distinct primes greater than 33. Each new terrace has at least one segment consisting of successive powers of 22, modulo nn.