New Sets of Optimal Odd-length Binary Z-Complementary Pairs

A pair of sequences is called a Z-complementary pair (ZCP) if it has zero aperiodic autocorrelation sums (AACSs) for time-shifts within a certain region, called zero correlation zone (ZCZ). Optimal odd-length binary ZCPs (OB-ZCPs) display closest correlation properties to Golay complementary pairs (GCPs) in that each OB-ZCP achieves maximum ZCZ of width (N + 1)/2 (where N is the sequence length) and every out-of-zone AACSs reaches the minimum magnitude value, i.e. 2. Till date, systematic constructions of optimal OB-ZCPs exist only for lengths 2α ± 1, where α is a positive integer. In this paper, we construct optimal OB-ZCPs of generic lengths 2α10β26γ + 1 (where α, β, γ are non-negative integers and α ≥ 1) from inserted versions of binary GCPs. The key leading to the proposed constructions is several newly identified structure properties of binary GCPs obtained from Turyn’s method. This key also allows us to construct OB-ZCPs with possible ZCZ widths of 4×10β-1+1, 12×26γ-1+1 and 12×10β26γ-1+1 through proper insertions of GCPs of lengths 10β, 26γ, and 10β26γ, respectively. Our proposed OB-ZCPs have applications in communications and radar (as an alternative to GCPs).