Abstract
A sequence a = (a(0), a(1), a(2), ... , a(n)) is said to be an almost p-ary sequence of period n + 1 if a(0) = 0 and a(i) = zeta(bi)(p) for 1 <= i <= n, where zeta(p) is a primitive p-th root of unity and b(i) epsilon {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 riot 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.