Almost p-ary Sequences

2018/07/30 by Özden, Büşra, Yayla, Oğuz
#05B10 #94A55 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1807.11412

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\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-γ₂ - 2)/(p)+γ₂,0,(n-γ₁ -1)/(p)+γ₁,(n-γ₂ - 2)/(p),(n-γ₁ -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.

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