Nearly perfect sequences with arbitrary out-of-phase autocorrelation

In this paper we study nearly perfect sequences (NPS) via their connection to direct product difference sets (DPDS). We prove the connection between a p-ary NPS of period n and type γ and a cyclic (n, p, n, n−γ p + γ, 0, n−γ p )-DPDS for an arbitrary integer γ. Next, we present the necessary conditions for the existence of a p-ary NPS of type γ. We apply this result for excluding the existence of some p-ary NPS of period n and type γ for n ≤ 100 and |γ| ≤ 2. We also prove the similar results for an almost p-ary NPS of type γ. Finally, we show the non-existence of some almost p-ary perfect sequences by showing the non-existence of equivalent cyclic relative difference sets by using the notion of multipliers.