Almost p-Ary Perfect Sequences

A sequence a = (a0, a1, a2, · · · , an) is said to be an almost p-ary sequence of period n + 1 if a0 = 0 and ai = ζ bi p for 1 ≤ i ≤ n, where ζp is a primitive p-th root of unity and bi ∈ {0, 1, · · · , p − 1}. Such a sequence a is called perfect if all its out-of-phase autocorrelation coefficients are zero; and is called nearly perfect if its out-of-phase autocorrelation coefficients are all 1, or are all −1. In this paper, on the one hand, we construct almost p-ary perfect and nearly perfect sequences; on the other hand, we present results to show they do not exist with certain periods. It is shown that almost p-ary perfect sequences correspond to certain relative difference sets, and almost p-ary nearly perfect sequences correspond to certain direct product difference sets. Finally, two tables of the existence status of such sequences with period less than 100 are given.