Nearly perfect sequences with arbitrary out-of-phase autocorrelation

Oǧuz  Yayla

doi:10.3934/amc.2016014

Article Contents

2016, Volume 10, Issue 2: 401-411. Doi: 10.3934/amc.2016014

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Nearly perfect sequences with arbitrary out-of-phase autocorrelation

Oǧuz Yayla^1,

1.
Department of Mathematics, Hacettepe University, Beytepe, 06800, Ankara

Received: August 31, 2014

Published: April 2016

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Abstract

A sequence of period $n$ is called a nearly perfect sequence of type $\gamma$ if all out-of-phase autocorrelation coefficients are a constant $\gamma$. 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.

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Mathematics Subject Classification: Primary: 05B10; Secondary: 94A55.

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