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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf,len=2170
+page_content='1  Two-Dimensional Golay Complementary Array  Sets With Arbitrary Lengths for  Omnidirectional MIMO Transmission  You-Qi Zhao, Cheng-Yu Pai, Zhen-Ming Huang, Zilong Liu, Senior  Member, IEEE, and Chao-Yu Chen, Member, IEEE  Abstract  This paper presents a coding approach for achieving omnidirectional transmission of certain common  signals in massive multi-input multi-output (MIMO) networks such that the received power at any  direction in a cell remains constant for any given distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Specifically, two-dimensional (2D) Golay  complementary array set (GCAS) can be used to design optimal massive MIMO precoding matrix so as  to achieve omnidirectional transmission due to its complementary autocorrelation property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' In this paper,  novel constructions of new 2D GCASs with arbitrary array lengths are proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Our key idea is to  carefully truncate the columns of certain larger arrays generated by 2D generalized Boolean functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Finally, the power radiation patterns and numerical results are provided to verify the omnidirectional  property of the GCAS-based precoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The error performances of the proposed precoding scheme are  presented to validate its superiority over the existing alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Index Terms  This work was supported by the Ministry of Science and Technology, Taiwan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=', under Grant MOST 109–2628–E–006–  008–MY3 and MOST 111–2218–E–305–002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  You-Qi Zhao and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Pai are with the Department of Engineering Science, National Cheng Kung University, Tainan 701,  Taiwan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' (e-mail: n98081505@gs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='ncku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='tw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Huang is with the Institute of Computer and Communication Engineering, National Cheng Kung University, Tainan  701, Taiwan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' (e-mail: n98101012@gs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='ncku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
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+page_content='tw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Zilong Liu is with the School of Computer Science and Electronic Engineering, University of Essex, United Kingdom (e-mail:  zilong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='liu@essex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Chen is with the Department of Electrical Engineering and the Institute of Computer and Communication Engineering,  National Cheng Kung University, Tainan 701, Taiwan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' (e-mail: super@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='ncku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='tw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  January 4, 2023  DRAFT  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='01225v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='IT]  3 Jan 2023  2  Generalized Boolean function (GBF), Golay complementary array pair (GCAP), Golay comple-  mentary array set (GCAS), omnidirectional precoding (OP), uniform rectangular array (URA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' INTRODUCTION  Complementary pairs/sets of sequences have attracted a sustained research interest owing to  their zero aperiodic correlation sums properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' To be specific, a Golay complementary pair  (GCP) refers to a pair of equal-length sequences whose summation of aperiodic autocorrelations  is zero except at the zero time-shift [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Such a concept was extended to Golay complementary  set (GCS) with constituent sequences of more than 2 by Tseng and Liu in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Furthermore, a  maximum collection of GCSs is called a set of complete complementary code (CCC) [3] if any  two different GCSs have zero aperiodic cross-correlation sums for all time-shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' In the literature,  GCSs and CCCs have been widely used for radar sensing [4], channel estimation [5], precoding  for massive multi-input multi-output (MIMO) [6], peak-to-average power ratio (PAPR) reduction  in orthogonal frequency division multiplexing (OFDM) [7]–[13], interference-free multicarrier  code division multiple access [14]–[17], and many other applications [18], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Recently, there is a surge of research attention to study two-dimensional (2D) Golay com-  plementary array sets (GCASs) [18]-[23], each having zero aperiodic autocorrelation sums  property for two directions of shifts (compared to conventional GCSs and CCCs with time-  shifts only).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' An important application of the 2D GCASs is for omnidirectional transmission in  MIMO communication systems with a uniform rectangular array (URA) configuration [20], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  In massive MIMO systems, some common messages (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=', reference signals, synchronization  signals, control signals, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=') need to be power-uniformly broadcasted to all the angles within  the whole cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' In this paper, we consider space-time block code (STBC) for the harvesting  of the diversity gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' At the base station (BS), the STBC encoded symbols are assigned to  several streams and then mapped onto the antenna arrays in URA by certain 2D GCASs assisted  precoding matrices to achieve uniform power radiation at any angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  On the other hand, since a large number of antennas are considered in massive MIMO  systems, a huge pilot overhead may be needed to acquire the channel state information (CSI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' As  pointed out in [22], this can be alleviated by omnidirectional precoding (OP) based transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  For uniform linear arrays (ULAs), Zadoff-Chu (ZC) sequences were adopted to satisfy the  requirements of the omnidirectional property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' However, [22] only considered the omnidirectional  January 4, 2023  DRAFT  3  transmission in certain directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Later in [6], GCSs and CCCs based OP matrices were proposed  to meet the requirement of omnidirectional transmission across all directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  In [20], [21], [23], [24], 2D GCASs were employed for precoding matrices in URAs by  applying interleaving and Kronecker-product to existing 1D sequences or 2D arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' As a result,  the array sizes of 2D GCASs are only feasible for certain lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' A construction of 2D GCASs  of array size pn × pm was proposed in [25] by using permutation ploynomials (PPs) functions  and 2-level autocorrelation sequences, where p is a prime number, m, n are two positive integers,  and p, m, n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Furthermore, a unifying construction framework for 2D GCASs was developed  in [26] by a multivariate polynomial matrix from certain seed para-unitary (PU) matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' In  [27], [28], Pai and Chen proposed direct constructions of 2D Golay complementary array pairs  (GCAPs) and GCASs with array size 2n × 2m from 2D generliazed Boolean functions (GBFs)  [29] where n, m are integers and n, m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 2D GCAP can be regarded as a case of 2D GCAS  when the set size is equal to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Moreover, Pai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' [30] proposed a direct construction of 2D  CCCs with array size 2n×2m, which have ideal autocorrelations and cross-correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Later, Liu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' [31] proposed a construction of GCASs with array size pn ×pm by using 2D multivariable  functions, where p is a prime number, n, m are integers, and n, m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Based on [27], [32]  developed a direct construction of GCASs with set size 4 and array size 2n × (2m−1 + 2v) by  using 2D GBFs, where n, m, v are positive number with n, m ≥ 2, and 0 ≤ v ≤ m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  The aforementioned research efforts are generally driven by the need of highly flexible array  sizes of 2D GCASs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Motivated by this, we aim for generating new GCASs with arbitrary array  lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The key idea of our proposed constructions is to carefully truncate some columns of  the certain larger arrays generated by 2D GBFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Thus, our proposed GCASs can be applied to  URAs with various array sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' In addition, the proposed GCASs can be directly generated from  2D GBFs without the requirements of any specific sequences or tedious sequence operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' In  Table I, we compare the existing parameters of 2D GCASs with our proposed ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  The remainder of this paper is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Section II discusses notations, definitions,  system models, and the omnidirectional transmission in MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Section III describes  our proposed constructions of 2D GCASs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Section IV shows the power radiation pattern and bit  error rate (BER) performance based on our proposed 2D GCASs precoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Finally, Section V  presents the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  January 4, 2023  DRAFT  4  TABLE I  A COMPARISON OF CONSTRUCTIONS FOR 2D GCASS  Construction  Parameters  Approaches  [26, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 5]  (N, N n, N m), N, n, m > 0  Seed PU matrices  [26, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 7]  (2k, 2kn, 2km), n, m, k > 0  [25, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 4]  (p, pn, pm), prime p, n, m > 0  PPs and 2-level  autocorrelation sequences  [25, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 6]  (pk, pkn, pkm), prime p, k, n, m > 0  [31, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 1]  (pk1  1 pk2  2 , pn  1 , pm  2 ), primes p1, p2  2D multivariable functions  [31, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 2]  (pk, pn, pm), prime p, n + m ≥ k > 0  [27], [28], [30]  (2k, 2n, 2m), n, m ≥ k > 0, and k > 0  2D GBFs  [32]  (4, 2n, 2m−1 + 2v), n, m ≥ 2, and k > 0  Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 1  (2k+1, 2n, 2m−1 + �k−1  α=1 dα2m−k+α−1 + d02v),  k < m, 0 ≤ v ≤ m − k, dα ∈ {0, 1}  Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 2  (2k+1, 2n, 2m−1 + �k−1  α=1 dα2π1(m−k+α)−1 + d02v),  k < m, 0 ≤ v ≤ m − k, dα ∈ {0, 1}  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' PRELIMINARIES AND DEFINITIONS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Notations  Throughout this paper, we present the notations in the following:  (a)i refers to the i-th element of the vector a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (A)i,j denotes the (i, j)-th element of the array A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (·)H refers to the conjugate transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  diag(A) refers to the column vector composed of the main diagonal of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (·)∗ refers to the complex conjugation of an element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (·)T refers to the transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  vec(·) express stacking one column of the matrix into one another column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  1 is a vector whose elements are all 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Let ξ = e2π√−1/q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  In this paper, q is an even number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Let X and Y be two arrays of size L1 × L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Then X and Y can be stated as  X = (Xg,i), Y = (Yg,i),  (1)  where g = 0, 1, · · · , L1 − 1 and i = 0, 1, · · · , L2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  January 4, 2023  DRAFT  5  Definition 1: Given two arrays X and Y of size L1 × L2, the 2D aperiodic cross-correlation  function (AACF) is defined by  ρ (X, Y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' u1, u2) =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  L1−1−u1  �  g=0  L2−1−u2  �  i=0  Yg+u1,i+u2X∗  g,i, 0 ≤ u1 < L1,  0 ≤ u2 < L2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  L1−1−u1  �  g=0  L2−1−u2  �  i=0  Yg+u1,iX∗  g,i−u2, 0 < u1 < L1,  −L2 < u2 < 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  L1−1−u1  �  g=0  L2−1−u2  �  i=0  Yg,iX∗  g−u1,i−u2, −L1 < u1 < 0,  −L2 < u2 < 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  L1−1+u1  �  g=0  L2−1−u2  �  i=0  Yg,i+u2X∗  g−u1,i, −L1 < u1 < 0,  0 < u2 < L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (2)  When X = Y , then it is called 2D aperiodic autocorrelation function (AACF) and denoted  by ρ(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' u1, u2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' If taking L1 = 1, two 2D arrays X and Y are degraded as a 1-D sequence  X = Xi for i = 0, 1, · · · , L2 − 1 and Y = Yi for i = 0, 1, · · · , L2 − 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Then the  1-D AACF of 1-D sequence X is related by  ρ(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' u) =  �  �  �  �  �  �  �  L2−1−u  �  i=0  Xi+uX∗  i ,  0 ≤ u ≤ L2 − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  L2−1+u  �  i=0  XiX∗  i−u,  −L2 + 1 ≤ u < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (3)  In this paper, q-PSK modulation is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Thus, x and y denote q-ary arrays and (1) is  expressed as  X = (Xg,i) = (ξxg,i) = ξx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Y = (Yg,i) = (ξyg,i) = ξy,  (4)  where x = (xg,i), y = (yg,i), and xg,i, yg,i ∈ Zq = {0, 1, · · · , q−1} for 0 ≤ g < L1, 0 ≤ i < L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Consider a set of N L-length sequences can be represented as  C = {X0, X1, · · · , XN−1}  where  Xn = (Xn,0, Xn,1, · · · , Xn,L−1)  for n = 0, 1, · · · , N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  January 4, 2023  DRAFT  6  Definition 2: [19] If a set C consisting of N sequences of length L satisfies  N−1  �  k=0  ρ(Xk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' u) =  �  �  �  �  �  NL,  u = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  0,  u ̸= 0,  (5)  then the set C is called a Golay complementary set of size N, denoted by (N, L)-GCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The  GCP can be regarded as a special case of the GCS by setting N = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Definition 3: For a GCP (X0, X1), if another GCP (Y0, Y1) meets the following condition:  ρ(X0, Y0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' u) + ρ(X1, Y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' u) = 0, for all u,  (6)  then the two GCPs are called the Golay complementary mate of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Definition 4: A pair of arrays X and Y of array size L1 × L2 is called a 2D Golay  complementary array pair if  ρ(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' u1, u2) + ρ(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' u1, u2) =  �  �  �  �  �  2L1L2,  u1 = u2 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  0,  u1 ̸= 0 or u2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (7)  Definition 5: Let the array set G = {X0, X1, · · · , XN−1} where each array in set G is of  size L1 × L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' If the array set G satisfies  N−1  �  k=0  ρ(Xk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' u1, u2) =  �  �  �  �  �  NL1L2,  u1 = u2 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  0,  u1 ̸= 0 or u2 ̸= 0,  (8)  the set G is called the Golay complementary array set of set size N denoted by (N, L1, L2)-  GCAS where L2 is defined as the length of the GCAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' If N = 2, the GCAS G is degraded as  a GCAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Generalized Boolean Functions  A 2D generalized Boolean function (GBF) f in n + m binary variables y1, y2, · · · , yn,  x1, x2, · · · , xm, is a function mapping: Zn  2 ×Zm  2 → Zq, where xi, yg ∈ {0, 1} for i = 1, 2, · · · , m  and g = 1, 2, · · · , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' A monomial of degree r is given by any product of r distinct variables  among y1, y2, · · · , yn, x1, x2, · · · , xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' For instance, x1x3y1y2 is a monomial of degree 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Next,  the variables z1, z2, · · · , zn+m are defined as  zl =  �  �  �  �  �  yl  if 1 ≤ l ≤ n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  xl−n  if n < l ≤ m + n,  (9)  January 4, 2023  DRAFT  7  which are useful for our proposed constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' For a 2D GBF with n + m variables, the 2D  Zq-valued array  f =  �  �  �  �  �  �  �  f0,0  f0,1  · ·  f0,2m−1  f1,0  f1,1  · ·  f1,2m−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  f2n−1,0  f2n−1,1  · ·  f2n−1,2m−1  �  �  �  �  �  �  �  (10)  of size 2n×2m is given by letting fg,i = f((g1, g2, · · · , gn), (i1, i2, · · · , im)), where (g1, g2, · · · , gn)  and (i1, i2, · · · , im) are binary vector representations of integers g = �n  h=1 gh2h−1 and i =  �n  j=1 ij2j−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Example 1: Taking q = 4, n = 2, and m = 3 for example, the 2D GBF is given as f =  3z5z4 + z2z3 + 2z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Then the array f of size 4 × 8 corresponding to f can be obtained, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=',  f =  �  �  �  �  �  �  �  0  0  0  0  0  0  3  3  0  0  0  2  1  1  3  3  2  3  2  3  2  3  1  2  2  3  2  3  2  3  1  2  �  �  �  �  �  �  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (11)  The GBF f can be rewritten as f = 3x3x2 + y2x1 + 2y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' In this paper, we consider the array  size ̸= 2n × 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Hence, we define the truncated array f (L) corresponding to the 2D GBF f by  ignoring the last 2m − L columns of the corresponding array f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Example 2: Following the same notations given in Example 1, the truncated array f (6) is  given by  f (6) =  �  ���  �  �  �  �  �  0  0  0  0  0  0  0  0  0  2  1  1  2  3  2  3  2  3  2  3  2  3  2  3  �  �  �  �  �  �  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (12)  For simplicity, we use f to stand for f (L) when L is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' System Model  Considering downlink transmission from a BS to UEs where each has one single antenna,  we suppose that the number of antennas at the BS is M = L1 × L2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=', the URA consists of  L1 rows and L2 columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 1 illustrates the diagram of data downlink transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' For an  January 4, 2023  DRAFT  8  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Diagram of data transmission through STBC encoding and omnidirectional precoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  L1 × L2 URA, the steering matrix A(ϕ, θ) at the direction (ϕ, θ) with the (g, i)-th entry can be  expressed as  (A(ϕ, θ))g,i =e−j 2π  λ gdy sin ϕ sin θ−j 2π  λ idx sin ϕ cos θ,  for g = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' , L1 − 1, i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' , L2 − 1,  θ ∈ [0, 2π], ϕ ∈ [0, π/2],  (13)  where dx and dy denote the vertical antenna and horizontal antenna inter-element spacings of  the URA, respectively, and λ denotes the carrier wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' To enhance the spatial diversity  and communication reliability, the STBC signal transmission scheme is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The N ×M STBC  is given by  S ≜  �  �  �  �  �  �  �  s0(0)  s0(1)  · ·  s0(M − 1)  s1(0)  s1(1)  · ·  s1(M − 1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  sN−1(0)  sN−1(1)  · ·  sN−1(M − 1)  �  �  �  �  �  �  �  ∈ CN×M  (14)  January 4, 2023  DRAFT  So(t)  Xo(t)  7  Data  Omni-  Space-time  UE1  Si(t)  directional  xi(t)  block coding  y(t)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  precoding  V  SN-1(t)  XL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='L2  1(t)  y  d  d  (t)  xo(t)  x  Omnidirectional  precoding  UE19  where CN×M refers to the N-by-M complex space and sn(t) denotes the (n, t)-th element of  the STBC at time instant t for t = 0, 1, · · · , M − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We define the precoding matrix Wn of size  L1 × L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The encoded symbols is given by  x(t) = (x0(t), x1(t), · · · , xL1L2−1(t))T = vec  �N−1  �  n=0  Wn · sn(t)  �  , for t = 0, 1, · · · , M − 1,  (15)  which are transmitted by the L1L2 antennas of the URA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' In the light-of-sight (LOS) channel  without multipaths, the received signal at the direction (ϕ, θ) can be written as  y(t) =  N−1  �  n=0  �  vec(A(ϕ, θ))Tvec(Wn)  �  sn(t) + η(t), t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=', M − 1,  (16)  where η(t) is the additive Gaussian white noise (AWGN) at time instant t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Omnidirectional Precoding Matrices Based on 2D Arrays  In this subsection, we list two necessary requirements for the design of OP matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Then,  we will connect these two requirements with the conditions of 2D arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Requirement 1 (R1): Omnidirectional transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  We consider the MIMO system with URA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Following (16), the received power E at the angle  (ϕ, θ) is represented as  E =  N−1  �  n=0  ��[vec(A(ϕ, θ))Tvec(Wn)]  ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (17)  Therefore, to satisfy the omnidirectional transmission in the whole cell, (17) must be constant  for all ϕ and θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Requirement 2 (R2): Equal average power on each antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  To enhance the efficiency of the power amplifier, the average transmission power on all L1×L2  antennas is required to be equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We define  W = (vec(W0), vec(W1), · · · , vec(WN−1)) ,  (18)  where the array size of W is L1L2 × N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Hence, (15) can be rewritten as  X = (x(0), x(1), · · · , x(M − 1)) = W S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (19)  January 4, 2023  DRAFT  10  Let s(t) be the t-th column of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Throughout this paper, we assume E  �  s(t)s(t)H�  =IN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The  transmitted signal on the (l1, l2)-th antenna is (W s)l2L1+l1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The average power on the (l1, l2)-th  antenna can be expressed as  E  �  |(W s)l2L1+l1|2�  =  �  W E  �  s(t)s(t)H�  W H�  l2L1+l1,l2L1+l1  = (W W H)l2L1+l1,l2L1+l1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (20)  Therefore, the condition to guarantee equal power on each antenna is equivalent to  diag(W W H) = N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (21)  Next, we will derive two sufficient conditions on the precoding matrices to fulfill requirements  R1 and R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Lemma 1: [21] For an L1 × L2 URA, if the precoding matrices W0, W1, · · · , WN−1 of size  L1 × L2 form an (N, L1, L2)-GCAS, then the omnidirectional transmission is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Lemma 2: For an L1×L2 URA, if the precoding matrices W0, W1, · · · , WN−1 of size L1×L2  are unimodular, then the average power on each antenna is equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Proof: In order to meet the requirement for equal average power on each antenna, the  precoding matrix W must satisfy (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We let wi = vec(Wi), for i = 0, 1, · · · , N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Then,  diag  �  W W H�  =  �N−1  �  i=0  |(wi)0|2 ,  N−1  �  i=0  |(wi)1|2 , · · · ,  N−1  �  i=0  |(wi)L1L2−1|2  �T  = N1  (22)  since we have  |(wi)n|2 = 1,  for i = 0, 1, · · · , N − 1 and n = 0, 1, · · · , L1L2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (23)  According to (21), the requirement (R2) is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  In the sequel, the design of OP matrices W0, W1, · · · , WN−1 are based on Lemma 1 and  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' That is, our goal is to construct unimodular GCASs with flexible sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' GCASS WITH FLEXIBLE ARRAY SIZE  In this section, two constructions of 2D GCASs with arbitrary array lengths based on 2D  GBFs will be proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' By recalling the function mapping in (9), we present our first theorem  in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  January 4, 2023  DRAFT  11  Theorem 1: For any integers q, m, n ≥ 2, and k < m, v is an integer satisfies 0 ≤ v ≤ m−k  and let π be a permutation of {1, 2, · · · m + n − k} satisfying {zπ(1), zπ(2), · · · , zπ(v+n)} =  {z1, z2, · · · , zv+n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The 2D generalized Boolean function can be written as  f = q  2  �m+n−k−1  �  l=1  zπ(l)zπ(l+1)  �  +  m+n  �  s=1  pszs + p0  (24)  where ps ∈ Zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The array set  G =  �  f + q  2  k  �  α=1  λαzm+n−k+α + q  2λk+1zπ(1) : λα ∈ {0, 1}  �  (25)  is a q-ary (2k+1, 2n, 2m−1 + �k−1  α=1 dα2m−k+α−1 + d02v)-GCAS where dα ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Proof: Without loss of generality, we consider L1 = 2n and L2 = 2m−1+�k−1  α=1 2m−k+α−1+ 2v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  We need to show that  �  c∈G  L1−1−u1  �  g=0  L2−1−u2  �  i=0  �  ξcg+u1,i+u2−cg,i�  = 0  (26)  for 0 ≤ u1 < 2n, 0 ≤ u2 < 2m−1 + �k−1  α=1 2m−k+α−1 + 2v and (u1, u2) ̸= (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Then we let  h = g + u1 and j = i + u2 for any integers g and i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We also let (g1, g2, · · · , gn),(i1, i2, · · · , im),  (h1, h2, · · · , hn), and (j1, j2, · · · , jm) be the binary representations of g, i, h, and j, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  For the ease of presentation, we denote  al =  �  �  �  �  �  gl  for 1 ≤ l ≤ n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  il−n for n < l ≤ n + m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  bl =  �  �  �  �  �  hl  for 1 ≤ l ≤ n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  jl−n for n < l ≤ n + m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (27)  In what follows, we consider four cases to show that the above formula holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Case 1: If aπ(1) ̸= bπ(1), we can find that c′ = c + (q/2)zπ(1) for any arrayc ∈ G satisfying  ch,j − cg,i − c′  h,j+c′  g,i = q  2(aπ(1) − bπ(1)) ≡ q  2  (mod q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (28)  Therefore, we have  ξch,j−cg,i + ξc′  h,j−c′  g,i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (29)  Case 2: If am+n−k+α ̸= bm+n−k+α, we can find that c′ = c + (q/2)zm+n−k+α for any array  c ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Similar to Case 1, we have  ξch,j−cg,i + ξc′  h,j−c′  g,i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (30)  January 4, 2023  DRAFT  12  Case 3: If aπ(1) = bπ(1) and am+n−k+α = bm+n−k+α for α = 1, 2, · · · , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Suppose that α′  is the largest integer satisfying am+n−k+α′ = bm+n−k+α′ = 0 for α′ ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Then we assume β  is the smallest integer which satisfies aπ(β) ̸= bπ(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Let a′ and b′ be integers distinct from a  and b, respectively, only in one position π(β − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' In other words, a′  π(β−1) = 1 − aπ(β−1) and  b′  π(β−1) = 1 − bπ(β−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' If 1 ≤ π(β − 1) ≤ n, by using the above definition, we have  cg′,i − cg,i  = q  2  �  aπ(β−2)g′  π(β−1) − aπ(β−2)gπ(β−1) + g′  π(β−1)aπ(β)  −gπ(β−1)aπ(β)  �  + pπ(β−1)g′  π2(β−1) − pπ(β−1)gπ(β−1)  ≡ q  2(aπ(β−2) + aπ(β)) + pπ(β−1)(1 − 2gπ(β−1))  (mod q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (31)  where a′  π(β−1) = g′  π(β−1) and aπ(β−1) = gπ(β−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Since aπ(β−2) = bπ(β−2) and aπ(β−1) = bπ(β−1),  we have  ch,j − cg,i − ch′,j + cg′,i  ≡ q  2(aπ(β−2) − bπ(β−2) + aπ(β) − bπ(β))  + pπ(β−1)(2hπ(β−1) − 2gπ(β−1))  ≡ q  2(aπ(β) − bπ(β)) ≡ q  2  (mod q)  (32)  implying ξch,j−cg,i/ξch′,j−cg′,i = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We can also obtain  ξch,j−cg,i + ξch′,j−cg′,i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (33)  If n < π(β − 1) ≤ n + m, note that a′  π(β−1) = i′  π(β−1)−n and aπ(β−1) = iπ(β − 1) − n according  to (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Following the similar argument as given above, we can get ξch,j−cg,i + ξch,j′−cg,i′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Case 4: If aπ(1) = bπ(1) and am+n−k+α = bm+n−k+α = 1 for α = 1, 2, · · · , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We assume β is  the smallest integer such that aπ(β) ̸= bπ(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Since as = bs = 0 for s = v+n+1, v+n+2, · · · , m+  n−k, we can obtain π(β) ≤ v+n implying π(β −1) ≤ v+n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' If 1 ≤ π(β−1) ≤ n, by following  the similar argument as given above, we have ξch,j−cg,i +ξch′,j−cg′,i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' If n < π(β −1) ≤ v+n,  we have ξch,j−cg,i + ξch,j′−cg,i′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' From Cases 1 to 4, the theorem can be proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Remark 1: The parameter 2m−1+�k−1  α=1 dα2m−k+α−1+d02v of the proposed GCASs in Theorem  1 can be any arbitrary length since m, k, v are flexible and dα ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Example 3: Taking q = 2, m = 6, n = 2, k = 1, and v = 0, we let π = (1, 2, 3, 4, 5, 6, 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  The generalized Boolean function is f = z1z2 + z2z3 + z3z4 + z4z5 + z5z6 + z6z7 = x1x2 +  x2x3 + x3x4 + x4x5 + y1y2 + y2x1 by setting pk = 0 for k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' , m + n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The array set  January 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 2023  DRAFT  13  TABLE II  THE CONSTRUCTED (4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 33)-GCAS IN EXAMPLE 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='c0 =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
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+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
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+page_content='  G = {f, f + x8, f + y1, f + x8 + y1} is a GCAS of size 4 and the array size is 4 × 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We let  G = {c0, c1, c2, c3} and list the constituent arrays in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 2 shows the AACF sum of  set G is zero at shift u1 ̸= 0 or u2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Thus, we can find that array set G is a (4, 4, 33)-GCAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  January 4, 2023  DRAFT  14  Next, we introduce a lemma which illustrates a construction of (4, 2n, 2m−1 +2v)-GCAS from  2D GBFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Lemma 3: [32, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 1] For nonnegative integers m, n, and v with 0 ≤ v < m − 1, let π1 be  a permutation of {1, 2, · · · , m − 1} and π2 be a permutation of {1, 2, · · · , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The 2D GBF is  given by  f =q  2  �m−2  �  k=1  xπ1(k)xπ1(k+1) +  n−1  �  k=1  yπ2(k)yπ2(k+1) + xπ1(m−1)xm + xmyπ2(1)  �  +  m  �  l=1  plxl +  n  �  s=1  κsys + p0  (34)  where pl, κs ∈ Zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Then the array set  G =  �  f, f + q  2xπ1(1), f + q  2yπ2(n), f + q  2xπ1(1) + q  2yπ2(n)  �  is a (4, 2n, 2m−1 + 2v)-GCAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Since the set size of the GCAS from Lemma 3 is limited to 4, we propose a general  construction of 2D GCASs with more flexible array sizes and set sizes which can include Lemma  3 as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Theorem 2: For any integers q, m, n ≥ 2, and k < m, v is an integer satisfies 0 ≤ v ≤ m−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Assume that π1 is a permutation of {1, 2, · · · m} and π2 is a permutation of {1, 2, · · · n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The  2D generalized Boolean function can be written as  f =q  2  �m−k−1  �  l=1  xπ1(l)xπ1(l+1) +  n−1  �  s=1  yπ2(s)yπ2(s+1) + xπ1(m)yπ2(n)  �  +  m−k  �  l=1  µlxπ1(l)xπ1(m) +  m  �  l=1  plxk +  n  �  s=1  κsys + p0  (35)  where µl, pl, κs, ∈ Zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The array set  G =  �  f + q  2  k−1  �  α=1  λαxπ1(m−k+α) + q  2λkyπ2(1) + q  2λk+1xπ1(1) : λα ∈ {0, 1}  �  (36)  is a q-ary (2k+1, 2n, 2m−1 + �k−1  α=1 dα2π1(m−k+α)−1 + d02v)-GCAS where dα ∈ {0, 1} if the  following three conditions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (C1) {π1(1), π1(2), · · · , π1(v)} = {1, 2, · · · , v} if v > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (C2) π1(m − k + α) < π1(m − k + α + 1) for 1 ≤ α ≤ k − 1 where π1(m) = m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (C3) For 1 ≤ α ≤ k − 1 and 2 ≤ β ≤ m − k, if π1(β) < π1(m − k + α), then π1(β − 1) <  π1(m − k + α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  January 4, 2023  DRAFT  15  Proof: Similarly, we consider L1 = 2n and L2 = 2m−1 + �k−1  α=1 2π1(m−k+α)−1 + 2v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Then  we would like to prove that  �  C  ρ(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' u1, u2) =  �  c∈G  L1−1−u1  �  g=0  L2−1−u2  �  i=0  �  ξcg+u1,i+u2−cg,i�  = 0  (37)  for 0 ≤ u1 < 2n, 0 ≤ u2 < 2m−1 + �k−1  α=1 2π1(m−k+α)−1 + 2v and (u1, u2) ̸= (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' From (4)  we can find that  c = q  2  �m−k−1  �  l=1  xπ1(l)xπ1(l+1) +  n−1  �  s=1  yπ2(s)yπ2(s+1) + xπ1(m)yπ2(n)  �  +  m−k  �  l=1  µlxπ1(l)xπ1(m) +  m  �  l=1  plxl +  n  �  s=1  κsys + p0 · 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (38)  Then we let h = g + u1 and j = i + u2 for any integers g and i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Next, we discuss seven cases  to complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Case 1: Assuming u1 > 0, u2 ≥ 0, and gπ2(1) ̸= hπ2(1), we can find an array c′ = c +  (q/2)yπ2(1) ∈ G for any array c ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Therefore, we can obtain  ch,j − cg,i − c′  h,j+c′  g,i = q  2(gπ2(1) − hπ2(1)) ≡ q  2  (mod q)  (39)  Since gπ2(1) ̸= hπ2(1), we have  ξch,j−cg,i/ξc′  h,j−c′  g,i = ξ  q  2 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (40)  Thus,  ξch,j−cg,i + ξc′  h,j−c′  g,i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (41)  Case 2: If u1 > 0, u2 ≥ 0, and gπ2(1) = hπ2(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Let β be the smallest integer such that  gπ2(β) ̸= hπ2(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We define g′ and h′ are two integers which are distinct from g and h only in  one position π2(β − 1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Then, similar to Case 2 of Theorem 1, we have  ξch,j−cg,i + ξch′,j−cg′,i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (42)  Case 3: We suppose im ̸= jm, u1 = 0 and u2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We let g′ be an integer distinct from  i only in one position, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=', g′  π2(n) = 1 − gπ2(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Similar to Case 3 of Theorem 1, we have  ξcg,j−cg,i + ξcg′,j−cg′,i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Case 4: If u1 = 0, u2 > 0, and iπ1(1) ̸= jπ1(1) or iπ1(m−k+α) ̸= jπ1(m−k+α), we can find an  array c′ = c + (q/2)xπ1(1) ∈ G or c′ = c + (q/2)xπ1(m−k+α) for any array c ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Similar to  Case 1, we can obtain ξcg,j−cg,i + ξc′  g,j−c′  g,i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  January 4, 2023  DRAFT  16  Case 5: Suppose u1 = 0, u2 > 0, iπ1(1) = jπ1(1), and iπ1(m−k+α) = jπ1(m−k+α) for all  α = 1, 2, · · · , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Suppose that α′ is the largest non-negative integer satisfying iπ1(m−k+α′) =  jπ1(m−k+α′) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Then we assume β is the smallest integer which satisfies iπ1(β) ̸= jπ1(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Here, we have is = js = 0 for s = π1(m − k + α′) + 1, π1(m − k + α′) + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' , m − 1, and  s ̸= π1(m − k + α) for α = α′ + 1, α′ + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Hence, it implies π1(β) < π1(m − k + α′)  and π1(β − 1) < π1(m − k + α′) according to the condition (C-3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Let i′ and j′ be integers that  differ from i and j, respectively, in the position π1(β − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Similar to Case 2, we have  ξcg,j−cg,i + ξcg,j′−cg,i′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (43)  Case 6: Suppose u1 = 0, u2 > 0, iπ1(1) = jπ1(1), and iπ1(m−k+α) = jπ1(m−k+α) = 1 for all  α = 1, 2, · · · , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Then we assume β is the smallest integer which satisfies iπ1(β) ̸= jπ1(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Since  is = js = 0 for s = v + 1, v + 2, · · · , m − k and s ̸= π1(m − k + α) for α = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' , k − 1, we  can obtain π1(β) ≤ v implying π1(β − 1) ≤ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Similar to Case 2, we have  ξcg,j−cg,i + ξcg,j′−cg,i′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (44)  From Cases 1 to 6, the theorem can be proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Remark 2: Taking σ2(l) = π2(n − l + 1) for l = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' , n and π1(m − k + α) = m − k + α  for α = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' , k in Theorem 2, (34) can be represented as  f =q  2  �m−k−1  �  k=1  xπ1(k)xπ1(k+1) +  n−1  �  k=1  yσ2(k)yσ2(k+1) + xmyσ2(1)  �  +  m−k  �  l=1  µlxπ1(l)xm  +  m  �  l=1  plxl +  n  �  s=1  κsys + p0  (45)  where pl, κs ∈ Zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We can find that the result of Lemma 3 is a special case of Theorem 2 by  simply setting k = 1, µm−1 = q  2, and µl = 0 for l = 1, · · · , m − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Example 4: Taking q = 2, m = 5, n = 2, k = 2, and v = 0, we let π1 = (1, 2, 4, 3, 5) and  π2 = (1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The generalized Boolean function is f = x1x2 + x2x4 + y1y2 + x5y1 by setting  pl, κs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The array set G is a GCAS of size 8 when the truncated size L1 = 4 L2 = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We  let G = {c0, c1, · · · , c7} and list the constituent arrays in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Also, their AACF sum is  shown as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' SIMULATION RESULTS  In this section, we present the numerical results including the power radiation pattern and  BER performance by using our proposed 2D GCASs for massive MIMO systems with URA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  January 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 2023  DRAFT  17  TABLE III  THE CONSTRUCTED (8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 21)-GCAS IN EXAMPLE 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='c0 =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
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+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
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+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Power Radiation Pattern  According to (16), the power radiation pattern �N−1  n=0  ��[vec(A(ϕ, θ))Tvec(Wn)]  ��2 can be ob-  tained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We first consider the massive MIMO system equipped with a URA of size 4 × 33,  January 4, 2023  DRAFT  18  0  200  2  400  40  600  20  800  0  1000  0  20  2  40  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The summation of autocorrelations of constituent arrays in the GCAS in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=', L1 = 4 and L2 = 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We take the GCS G = {c0, c1, c2, c3} listed in Table II to  generate the precoding matrices {W0, W1, W2, W3} = {(−1)c0, (−1)c1, (−1)c2, (−1)c3} with  the omnidirectional property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The power radiation pattern of the GCAS-based scheme with array  size 4 × 33 is perfectly omnidirectional as illustrated in Fig 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  For the purpose of comparison, we also show the power radiation patterns of the precoding  matrices based on Zadoff-Chu sequences and random-matrices whose elements are randomly  generated from “+1” and “−1”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The ZC-based precoder consists of four 4 × 33 precoding  matrices, which are obtained based on a ZC sequence of length 4 and a ZC sequence of length  33 [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 4(b) illustrates the power radiation pattern of the ZC-based precoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We can  find that its power radiation pattern is not omnidirectional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The random-matrix-based precoder  consists of four 4 × 33 precoding matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The elements in the random-matrix-based precoding  matrices are generated by selecting the elements from {1, −1} with equal probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 4(c)  describes the power radiation pattern of the random matrix-based precoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We can observe that  the power radiation pattern is not omnidirectional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Next, we consider the massive MIMO system equipped with a URA of size 4 × 21, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=',  January 4, 2023  DRAFT  19  (a) GCAS-based precoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (b) ZC-based precoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (c) Random-matrix-based precoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Power radiation pattern with 4 × 33 URA and 4 × 4 STBC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  L1 = 4 and L2 = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We use the GCS G = {c0, c1, · · · , c7} listed in Table III for the precoding  matrix {W0, W1, · · · , W7} = {(−1)c0, (−1)c1, · · · , (−1)c7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The power radiation pattern of the  GCAS-based scheme with array size 4×21 is described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The perfect omnidirectional  property can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We also see that the power radiation patterns of the ZC-based precoder  and the random-matrix precoder shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 5(b) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 5(c) are not omnidirectional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The  ZC-based precoding matrices are obtained by a ZC sequence of length 4 and ZC sequence of  21 [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Bit Error Rate Performance  In this subsection, we present the BER performance of our proposed 2D GCAS-based schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  We first consider the massive MIMO system equipped with a URA of size 4×33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We let N = 4  January 4, 2023  DRAFT  X-axis-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='51Sy-ax1sX-axis550.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='51Sy-ax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  1S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='3X-axis00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='5Sy-ax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  1S20  (a) GCAS-based precoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (b) ZC-based precoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (c) Random-matrix-based precoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Power radiation pattern with 4 × 21 URA and 8 × 8 STBC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  and then the 4 × 4 orthogonal real STBC be presented as  S =  �  �  �  �  �  �  �  s0  −s1  −s2  −s3  s1  s0  s3  −s2  s2  −s3  s0  s1  s3  s2  −s1  s0  �  �  �  �  �  �  �  ,  (46)  where s0, s1, s2, s3 are binary phase shift keying (BPSK) modulated symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We employ the  maximum likelihood (ML) decoding here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' For each realization, the elevation and the azimuth  angles are uniformly distributed at random between [0, π/2] and [0, 2π], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' For com-  parison, the ZC-based precoder and random-matrix-based precoder are the same as mentioned in  Section IV-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The BER performances of three different schemes are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We can  find that the 2D GCAS-based scheme outperform the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' At BER of 10−4, there are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='6 dB  and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='6 dB gains over the ZC-based scheme and the random-matrix-based scheme, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  January 4, 2023  DRAFT  X-axis-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='51Sy-ax1sX-axis-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='51Sy-ax1s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='8X-axis-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='2Sy-06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='ax1s21  2  0  2  4  6  8  10  12  14  SNR (dB)  10-8  10-6  10-4  10-2  100  BER  GCAS-based precoding (N=4)  ZC-based precoding (N=4)  Random-matrix-based precoding (N=4)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' BER performance of the different schemes for a 4 × 33 URA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Next, we consider the massive MIMO system equipped with a URA of size 4 × 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We  consider 8 × 8 STBC and the 8 × 8 orthogonal real STBC is given by  S =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  s0  s1  s2  s3  s4  s5  s6  s7  −s1  s0  s3  −s2  s5  −s4  −s7  s6  −s2  −s3  s0  s1  s6  s7  −s4  −s5  −s3  s2  −s1  s0  s7  −s6  s5  −s4  −s4  −s5  −s6  −s7  s0  s1  s2  s3  −s5  s4  −s7  s6  −s1  s0  −s3  s2  −s6  s7  s4  −s5  −s2  s3  s0  −s1  −s7  −s6  s5  −s4  −s3  s2  s1  s0  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  (47)  where s0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' s1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' s7 are BPSK modulated symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We also take the ZC-based precoding and  random-matrix-based precoding for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The BER performance comparison for these  three different schemes is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' At BER of 10−4, there are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='2 dB and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='8 dB  gains over the ZC-based scheme and the random-matrix-based scheme, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' As a result,  the 2D GCASs are good candidates as precoding matrices for omnidirectional transmission in  January 4, 2023  DRAFT  22  2  0  2  4  6  8  10  12  14  SNR (dB)  10-8  10-6  10-4  10-2  100  BER  GCAS-based precoding (N=8)  ZC-based precoding (N=8)  Random-matrix-based precoding (N=8)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' BER performance of the different schemes for a 4 × 21 URA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  massive MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' CONCLUSION  In this paper, constructions of 2D GCASs with flexible array sizes have been proposed in  Theorems 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Our constructions can be obtained directly from 2D GBFs without the aid  of special sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Besides, our proposed GCASs have flexible array sizes which can fit  more antenna configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Furthermore, Theorem 2 can include the results in [32] as a special  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Simulation results showed that the omnidirectional transmission can be achieved when  the precoding matrices are based on the proposed GCASs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The BER performance due to their  omnidirectional power radiation patterns, the ZC-based scheme and random-matix-based have  inferior performances because their power radiation patterns both are not ideally omnidirectional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Although Theorems 1 and 2 can provide direct constructions of 2D GCASs, the first dimension  has size L1 limited to 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Therefore, the future work includes the extension of constructions of  2D GCASs of which both dimensions have non-power-of-two sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
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