Almost p-ary Sequences

In this paper we study almost $p$-ary sequences and their autocorrelation coefficients. We first study the number $\ell$ 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 $\ell$. It is shown that $\ell$ 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 $(n+2,p,n,\frac{n-\gamma_2 - 2}{p}+\gamma_2,0,\frac{n-\gamma_1 -1}{p}+\gamma_1,\frac{n-\gamma_2 - 2}{p},\frac{n-\gamma_1 -1}{p})$ 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 don't exist for $\gamma_2 \leq -3$.