On (n,k)-sequences

An (n,k)-sequence has been studied. A permutation a₁,a₂,...,a_kn of 0,1,...,kn-1 is an (n,k)-sequence if a_s+d-a_s≢a_t+d-a_t(modn) whenever ⌊a_s+d/n⌋=⌊a_s/n⌋ and ⌊a_t+d/n⌋=⌊a_t/n⌋ for every s,t and d with 1≤s<t<t+d≤kn, where ⌊x⌋ is the integer part of x. We recall the "prime construction" of an (n,k)-sequence using a primitive root modulo p whenever kn+1=p is an odd prime. In this paper we show that (n,k)-sequences from the prime construction for a given p are "essentially the same" with each other regardless of the choice of primitive roots modulo p. Further, we study some interesting properties of (n,k)-sequences, especially those from prime construction. Finally, we present an updated table of essentially distinct (n,2)-sequences for n≤13. The smallest n for which the existence of an (n,2)-sequences is open now becomes 16.