Optimal Odd-Length Binary Z-Complementary Pairs

Abstract

A pair of sequences is called a Golay complementary pair (GCP) if their aperiodic auto-correlation sums are zero for all out-of-phase time shifts. Existing known binary GCPs only have even-lengths in the form of 2 α 10 β 26 γ (where α, β, γ are non-negative integers). To fill the gap left by the odd-lengths, we investigate the optimal odd-length binary pairs which display the closest correlation property to that of GCPs. Our criteria of "closeness" is that each pair has the maximum possible zerocorrelation zone (ZCZ) width and minimum possible out-of-zone aperiodic auto-correlation sums. Such optimal pairs are called optimal odd-length binary 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 (ADF) which are useful in studying partially balanced incomplete block design (BIBD).

Single-Carrier SM-MIMO: A Promising Design for Broadband Large-Scale Antenna Systems

“…The poten-153 tial advantages and disadvantages of the SC-SM scheme are 154 summarized in Section III. In Section IV the applications of 155 the SC-SM transceiver to LS-MIMO systems are presented, 156 where different SC-SM designs conceived for the multiuser LS- 157 MIMO uplink and downlink are reviewed. Section V highlights 158 information bits, which will be transmitted by the activated 191 TA.…”

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

“…Motivated by optimal OB-ZCPs of length 2 α + 1 obtained from inserted versions of binary GCPs of length 2 α , we aim at systematic constructions of optimal OB-ZCPs with generic lengths of 2 α 10 β 26 γ + 1. Unlike the binary GCPs used in [12] which are constructed from generalized Boolean functions (and thus have lengths of 2 α only) proposed by Davis and Jedwab [5], we use Turyn's method [27] to generate binary GCPs of lengths 2 α 10 β 26 γ . Several intrinsic structure properties of the GCPs from Turyn's method are identified, for the first time.…”

“…Definition 5 (Optimal OB-ZCP [12]): An OB-ZCP (Type-I or Type-II) is said to be optimal if it is Z-optimal and every out-of-zone AACS takes the minimum magnitude value of 2.…”

“…[12] Every Type-I (or Type-II) OB-ZCP of length N (odd) has a maximum ZCZ width (N + 1)/2, i.e., Z ≤ (N + 1)/2.…”

Good Code Sets from Complementary Pairs via Discrete Frequency Chips

“…Galati et al [13] also discusses in detail this effect of unequal attenuation on the complementary code pairs. Liu et al [14] describes z-complementary code pairs that achieve a zero-correlation zone around the mainlobe and minimum possible sidelobe peaks outsize the zero domain. However, these codes also require two frequencies or channels which makes them vulnerable to the aforementioned effects of fading.…”