Nearly perfect sequences with arbitrary out-of-phase autocorrelation

TL;DR

This paper explores the properties and existence conditions of nearly perfect sequences with arbitrary out-of-phase autocorrelation, establishing connections to difference sets and providing non-existence results for certain parameters.

Contribution

It introduces a new connection between nearly perfect sequences and difference sets, and derives necessary conditions for their existence, including non-existence proofs for specific cases.

Findings

Established a link between p-ary NPS and cyclic difference sets.

Derived necessary conditions for the existence of p-ary NPS.

Proved non-existence of certain almost p-ary perfect sequences for specific parameters.

Abstract

In this paper we study nearly perfect sequences (NPS) via their connection to direct product difference sets (DPDS). We prove the connection between a $p$-ary NPS of period $n$ and type $\gamma$ and a cyclic $(n,p,n,\frac{n-\gamma}{p}+\gamma,0,\frac{n-\gamma}{p})$-DPDS for an arbitrary integer $\gamma$. Next, we present the necessary conditions for the existence of a $p$-ary NPS of type $\gamma$. We apply this result for excluding the existence of some $p$-ary NPS of period $n$ and type $\gamma$ for $n \leq 100$ and $\vert \gamma \vert \leq 2$. We also prove the similar results for an almost $p$-ary NPS of type $\gamma$. Finally, we show the non-existence of some almost $p$-ary perfect sequences by showing the non-existence of equivalent cyclic relative difference sets by using the notion of multipliers.