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Nearly perfect sequences with arbitrary out-of-phase autocorrelation. (English) Zbl 1348.94024

Summary: 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 \(| \gamma | \leq 2\). 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.

MSC:

94A55 Shift register sequences and sequences over finite alphabets in information and communication theory
05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)
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