This paper is devoted to sequences and focuses on designing new two-dimensional (2-D) Z-complementary array pairs (ZCAPs) by exploring two promising approaches. A ZCAP is a pair of 2-D arrays, whose 2-D autocorrelation sum gives zero value at all time shifts in a zone around the $(0,0)$ time shift, except the $(0,0)$ time shift. The first approach investigated in this paper uses a one-dimensional (1-D) Z-complementary pair (ZCP), which is an extension of the 1-D Golay complementary pair (GCP) where the autocorrelations of constituent sequences are complementary within a zero correlation zone (ZCZ). The second approach involves directly generalized Boolean functions (which are important components with many applications, particularly in (symmetric) cryptography). Along with this paper, new construction of 2-D ZCAPs is proposed based on 1-D ZCP, and direct construction of 2-D ZCAPs is also offered directly by 2-D generalized Boolean functions. Compared to existing constructions based on generalized Boolean functions, our proposed construction covers all of them. ZCZ sequences are a class of spreading sequences having ideal auto-correlation and cross-correlation in a zone around the origin. In recent years, they have been extensively studied due to their crucial applications, particularly in quasi-synchronous code division multiple access systems. Our proposed 2-D ZCAPs based on 2-D generalized Boolean functions have larger 2-D $\mathrm{ZCZ}_{\mathrm{ratio}}=\frac{6}{7}$. Compared to the construction based on ZCPs, our proposed 2-D ZCAPs also have the largest 2-D $\mathrm{ZCZ}_{\mathrm{ratio}}$.
Generalized Constructions of Complementary Sets of Sequences of Lengths Non-Power-of-Two
由于OFDM系统的实际应用，具有不同集合大小和序列长度的序列的互补集合（CS）的构造变得重要。基于广义布尔函数（GBF）的CS的大多数构造的长度为$2^α$（$α$是自然数）。最近，已经报道了一些关于构造具有非2的幂的长度的CS的工作，即，形式为$2^{m-1}+2^v$（$m$是自然数，$0\leq v <m $）、$N+1$和$N+2$，其中$N$是$q$进制互补对存在的长度。在本文中，我们提出了一个建设的CS长度$M+N$为集大小$4n$，使用级联的CS长度$M$和$N$，和集大小$4n$，其中$M$和$N$是长度$q$元互补对存在。此外，我们通过连接长度为$M$和$P$的CS以及集合大小为$8n$的CS来构造集合大小为$8n$的长度为$M+P$的CS，其中$M$和$P$分别是存在$q$进制互补对和大小为$4$的互补集合的长度。所提出的结构涵盖了所有以前的结构作为特殊情况下的长度，并导致更多的CS的新的序列长度，这是以前没有报道过的。
Non-standard Golay Complementary Sequence Pair over QAM
我们将Frank Fiedler等人在2008年给出的用于构造和枚举Golay阵列和序列对的三阶段过程[A multi-dimensional approach to the construction and enumeration of Golay complementary sequences，Journal of Combinatorial Theory，Series A 115（2008）753-776]推广到基于仿酉矩阵方法的$4^{q}$ -QAM星座，这在一定程度上解决了他们悬而未决的问题。我们的工作不仅包含了$4^{q}$ -QAM上基于布尔函数的Golay互补序列和QPSK上标准Golay序列偶的已知结果的主要部分，而且基于QPSK上非标准Golay阵列偶生成了新的$4^{q}$ -QAM上的Golay互补阵列（序列）。
Constructions of Polyphase Golay Complementary Arrays
Key Laboratory for Information Science of Electromagnetic Waves (MoE), Department of Communication Science and Engineering, School of Information Science and Technology, Fudan University（复旦大学信息科学与技术学院通信科学与工程系电磁波信息科学教育部重点实验室）
