Almost p-ary sequences

Abstract

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 $\min \limits \{s,p,n\}$. 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 $(n+2,p,n,\frac {n-\gamma _{2} - 2}{p}+\gamma _{2},0,\frac {n-\gamma _{1} -1}{p}+\gamma _{1},\frac {n-\gamma _{2} - 2}{p},\frac {n-\gamma _{1} -1}{p})$ PDPDS for arbitrary integers γ₁ and γ₂. Then we prove a necessary condition on γ₂ for the existence of such sequences. In particular, we show that they do not exist for γ₂ ≤ − 3.