In this paper we study almost p-ary sequences and their autocorrelation coefficients. We first study the number ℓ of distinct out-of-phase autocorrelation coefficients for an almost p-ary sequence of period n + s with s consecutive zero-symbols. We prove an upper bound and a lower bound on ℓ. It is shown that ℓ can not be less than min{s, p, n}. In particular, it is shown that a nearly perfect sequence with at least two consecutive zero symbols does not exist. Next we define a new difference set, partial direct product difference set (PDPDS), and we prove the connection between an almost p-ary nearly perfect sequence of type (γ1, γ2) and period n + 2 with two consecutive zero-symbols and a cyclic (n + 2, p, n, n−γ 2 −2 p + γ2, 0, n−γ 1 −1 p + γ1, n−γ 2 −2 p , n−γ 1 −1 p ) PDPDS for arbitrary integers γ1 and γ2. Then we prove a necessary condition on γ2 for the existence of such sequences. In particular, we show that they don't exist for γ2 ≤ −3.

In this paper, we study the m-ary sequences with (non-consecutive) two zero-symbols and at most two distinct autocorrelation coefficients, which are known as almost m-ary nearly perfect sequences. We show that these sequences are equivalent to -partial direct product difference sets (PDPDS), then we extend known results on the sequences with two consecutive zerosymbols to non-consecutive case. Next, we study the notion of multipliers and orbit combination for -PDPDS. Finally, we present two construction methods for a family of almost quaternary sequences with at most two out-of-phase autocorrelation coefficients.