Optimal Odd-Length Binary Z-Complementary Pairs

Abstract

A pair of sequences is called a Golay complementary pair (GCP) if their aperiodic autocorrelation sums are zero for all out-of-phase time shifts. Existing known binary GCPs only have even-lengths in the form of 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</sup> 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">β</sup> 26 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">γ</sup> (where \(α, β, γ) are nonnegative integers). To fill the gap left by the odd-lengths, we investigate the optimal odd-length binary (OB) pairs, which display the closest correlation property to that of GCPs. Our criteria of closeness is that each pair has the maximum possible zero-correlation zone (ZCZ) width and minimum possible out-of-zone aperiodic autocorrelation sums. Such optimal pairs are called optimal OB Z-complementary pairs (OB-ZCP) in this paper. We show that each optimal OB-ZCP has maximum ZCZ width of (N+1)/2, and minimum out-of-zone aperiodic sum magnitude of 2, where N denotes the sequence length (odd). Systematic constructions of such optimal OP-ZCPs are proposed by insertion and deletion of certain binary GCPs, which settle the 2011 Li-Fan-Tang-Tu open problem positively. The proposed optimal OB-ZCPs may serve as a replacement for GCPs in many engineering applications, where odd sequence lengths are preferred. In addition, they give rise to a new family of base-two almost difference families, which are useful in studying partially balanced incomplete block design.