Optimal and Z-Optimal Type-II Odd-Length Binary Z-Complementary Pairs

A pair of sequences is referred to as a type-II Z-complementary pair (ZCP) if their out-of-phase aperiodic autocorrelation sums (AACSs) equal zeros for time-shifts around the end-shift position. Specially, a type-II ZCP degenerates to a Golay complementary pair (GCP) if the aforementioned AACSs vanish for all nonzero time shifts. By generalizing insertion and deletion functions, systematic constructions are proposed so as to convert an arbitrary binary GCP into a type-II odd-length or even-length binary ZCP (OB-ZCP/EB-ZCP), depending on the number of entries of insertion or deletion. By insertion construction, further, existence of type-II binary ZCPs for arbitrary length larger than 2 is verified. In particular, by inserting one entry in an arbitrary or a carefully-chosen binary GCP of even length $N$ , the resultant type-II OB-ZCP of length N+1 is Z-optimal or optimal. Similarly, so is the resultant type-II OB-ZCP of length N-1 for deleting one entry. The proposed pairs provide more flexible choices for applications.