Almost p-ary sequences

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 (gamma(1), gamma(2)) and period n + 2 with two consecutive zero-symbols and a cyclic (n+2,p,n,n-gamma 2-2p+gamma 2,0,n-gamma 1-1p+gamma 1,n-gamma 2-2p,n-gamma 1-1p) PDPDS for arbitrary integers gamma(1) and gamma(2). Then we prove a necessary condition on gamma(2) for the existence of such sequences. In particular, we show that they do not exist for gamma(2) <= - 3.