Optimal binary periodic almost-complementary Pairs

A pair of sequences is called a periodic complementary pair (PCP) if the periodic autocorrelations of the constituent sequences sum up to zero for all nonzero time shifts. Owing to the scarcity of PCPs, we investigate optimal binary periodic almost-complementary pairs (BP-ACPs), each displaying correlation property closest to that of PCP. We show that an optimal BP-ACP of even length N has zero out-of-phase periodic autocorrelation sums (PACSs) except at the time shift of N/2, where the corresponding PACS has minimum magnitude of 4. We also show that for any arbitrary odd N, all the out-of-phase PACSs of an optimal BP-ACP should have identical magnitude of 2. A number of optimal BP-ACPs from analytical constructions as well as computer search are presented. In addition, our proposed optimal BP-ACPs for the even-length case lead to two new families of base-two almost difference families.