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 $2^{\alpha } \pm 1$ , where $\alpha $ is a positive integer. In this paper, we construct optimal OB-ZCPs of generic lengths $2^\alpha 10^\beta 26^\gamma +1$ (where $\alpha,~\beta,~\gamma $ are non-negative integers and $\alpha \geq 1$ ) 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 $4 \times 10^{\beta -1} +1$ , $12 \times 26^{\gamma -1}+1$ and $12 \times 10^\beta 26^{\gamma -1}+1$ through proper insertions of GCPs of lengths $10^\beta,~26^\gamma, \text {and } 10^\beta 26^\gamma $ , respectively. Our proposed OB-ZCPs have applications in communications and radar (as an alternative to GCPs).

中文翻译：

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新的最优奇数长度二进制 Z 互补对集

如果一对序列在特定区域（称为零相关区（ZCZ））内的时移具有零非周期自相关和（AACS），则称为 Z 互补对（ZCP）。 最佳 奇数长度二进制 ZCP (OB-ZCP) 显示与 Golay 互补对 (GCP) 最接近的相关属性，因为每个 OB-ZCP 实现了最大宽度 ZCZ $({N}+1)/2$ （其中 N 是序列长度）并且每个区外 AACS 达到最小幅度值，即 2。迄今为止，系统构建 最佳 OB-ZCP 仅存在于长度 $2^{\alpha } \pm 1$ ， 在哪里 $\alpha $ 是一个正整数。在本文中，我们构造最佳 OB-ZCPs 通用的 长度 $2^\alpha 10^\beta 26^\gamma +1$ （在哪里 $\alpha,~\beta,~\gamma $ 是非负整数并且 $\alpha\geq 1$ ) 来自插入版本的二进制 GCP。这钥匙导致提出的结构是从Turyn方法获得的二元GCP的几个新确定的结构特性。这钥匙 还允许我们构建具有可能的 ZCZ 宽度的 OB-ZCP $4 \times 10^{\beta -1} +1$ , $12 \times 26^{\gamma -1}+1$ 和 $12 \times 10^\beta 26^{\gamma -1}+1$ 通过适当插入长度的 GCP $10^\beta,~26^\gamma,\text {and} 10^\beta 26^\gamma $ ， 分别。我们提出的 OB-ZCP 可用于通信和雷达（作为 GCP 的替代品）。