diff --git "a/3dA0T4oBgHgl3EQfNP-W/content/tmp_files/load_file.txt" "b/3dA0T4oBgHgl3EQfNP-W/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/3dA0T4oBgHgl3EQfNP-W/content/tmp_files/load_file.txt" @@ -0,0 +1,595 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf,len=594 +page_content='1 A Direct Construction of Near-Optimal Multiple ZCZ Sequence Sets Nishant Kumar, Sudhan Majhi, Senior Member, IEEE, and Ashish K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Upadhyay Abstract—In this paper, for the first time, we present a direct construction of multiple zero-correlation zone (ZCZ) sequence sets with inter-set zero-cross correlation zone (ZCCZ) from generalised Boolean function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' The presented ZCZ sequence sets are optimal and their union is near-optimal ZCZ sequence set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' This work partially settles the open problem introduced by Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' in their 2010 paper using direct construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' The proposed construction is presented by two layer graphical representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Finally, the construction is compared with existing state-of-the- art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Index Terms—Generalised Boolean function (GBF), zero-cross correlation zone (ZCCZ), zero-correlation zone (ZCZ), multiple ZCZ sequence sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' INTRODUCTION Z complementary pairs (ZCPs) were introduced by Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' [1] to overcome the limitation on the lengths of Golay complementary pairs (GCPs) [2]–[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' The idea of ZCPs was generalized to Z-complementary code set (ZCCS) by Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' A ZCCS refers to a set of K codes, each of which consists of M constituent sequences of identical length L, having ideal aperiodic auto- and cross-correlation properties inside the ZCZ width (Z) [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' When Z = L and K = M, the set is called complete complementary code CCC [9]– [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' To reduce the “near-far effect” and ensure interference- free communication in asynchronous CDMA systems, ZCZ sequences were introduced in the late 1990s [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' When the received signal delays within ZCZ, ZCZ sequences can be employed to remove or reduce MAI and multipath interference (MPI) in quasi synchronous CDMA (QS-CDMA) systems [13], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Although the ZCZ spreading sequences prevent co- channel interference within each cell, inter-cell interference across neighbouring cells is unavoidable [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' To address the aforementioned shortcoming, the idea of multiple ZCZ sequence sets with inter set zero-cross corre- lation zone (ZCCZ) has recently been proposed [16]–[23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' A multiple ZCZ sequence set comprises ZCZ sequence sets as its subsets and the cross-correlation function between two arbitrary sequences from different subsets has either ZCCZ or low cross-correlation zone (LCCZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Authors in [24] and [25] used generalised bent function and perfectly non-linear functions respectively to construct multiple ZCZ sequence sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' But they tend to achieve only multiple ZCZ sequence set with interset LCCZ instead of ZCCZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' In [26], authors presented construction of multiple ZCZ sequence sets using discrete Fourier transform (DFT) matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Furthermore, an asymmetric ZCZ (A-ZCZ) sequence set is a multiple ZCZ sequence set and the ZCCZ between two arbitrary sequences from distinct subsets has a large ZCCZ [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' To obtain A-ZCZ sequence sets, interleaving techniques on perfect sequences are presented in the literature [21]–[23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Since, perfect se- quences are available only for very few lengths therefore these constructions also have very limited lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Additionally, the DFT matrices [18]–[20] and Hadamard matrices [16] are also used to construct A-ZCZ sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' But, all these constructions are indirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' The limitation of A-ZCZ sequence set is that the large ZCCZ is obtained at the cost of optimality of ZCZ sequence sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' In [17], Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' proposed a method for constructing multiple binary ZCZ sequence sets from mutually orthog- onal Golay complementary set (MOGCS) with good inter- set cross-correlation property and provided an open problem as* “we propose the following open problem: Construct N ZCZ sequence sets Zi, 0 ≤ i < N, satisfy: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Each Zi is an (K, Z, L)-ZCZ sequence set with KZ/L = 1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' The sets have a common zero correlation zone of length Zc with Zc = Z/N”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Motivated by the above open problem given in [17], in this letter, we propose a direct construction of near-optimal mul- tiple ZCZ sequence sets using generalised Boolean function (GBF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Since, proposed construction is based on GBFs, there- fore it is suitable for rapid hardware generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' A graphical analysis of our proposed construction has also been provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Also, it is the first time that a direct construction of multiple ZCZ sequence sets with ZCCZ is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' The proposed construction generalizes construction given in [13] and it is optimal over several constructions of A-ZCZ sequence sets presented in [16], [18]–[23], [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' NOTATIONS AND DEFINITIONS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' De��nition and Correlation Functions Let a1 = (a10, a11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , a1(L−1)) and a2 = (a20, a21, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , a2(L−1)) be two sequences of equal length L, having entries from complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' For an integer u, we define aperiodic cross-correlation function (ACCF) of sequences a1 and a2 as γ(a1, a2)(u) = ��L−1−u i=0 a1ia∗ 2(i+u), 0 ≤ u < L, �L+u−1 i=0 a1(i−u)a∗ 2i, −L < u < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' (1) Moreover, ACCF is termed as aperiodic auto-correlation func- tion (AACF) if a1 = a2 and denoted as γ(a1)(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Next, we define periodic cross-correlation function (PCCF) in terms of ACCF as φ(a1, a2)(u) = γ(a1, a2)(u) + γ∗(a2, a1)(L − u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' (2) The notations has been changed as per this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='02144v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='IT] 5 Jan 2023 2 Definition 1: Let C = {C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , CP −1} be a collection of P codes (matrices) having M rows and L columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Define Cη = [aη 0 aη 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' aη M−1]T M×L, (3) where aη ν (0 ≤ ν ≤ M − 1, 0 ≤ η ≤ P − 1) is the νth row sequence or νth constituent sequence and [·]T represents trans- pose of matrix [·].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Then the ACCF of two codes Cη1, Cη2 ∈ C is defined as γ(Cη1, Cη2)(u) = M−1 � ν=0 γ(aη1 ν , aη2 ν )(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' (4) Definition 2: Let C be a code set as defined in (3) which satisfies following correlation properties γ(Cη1, Cη2)(u) = � � � � � LM, η1 = η2 and u = 0, 0, η1 = η2 and 0 < |u| < L, 0, η1 ̸= η2 and |u| < L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' (5) Then C is known as (P, M, L)-MOGCS and each code in C is called GCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Moreover, if P = M then C is known as CCC set and denoted by (P, P, L)-CCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Definition 3: Let Zl = {zl 0, zl 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , zl K−1} be a collection of K L-length sequences, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=', zl i = (zl i0, zl i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , zl iL−1), 0 ≤ i ≤ K − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Then, Z is called (K, Z, L)-ZCZ sequence set if it satisfies following, φ(zl i, zl j)(u) = � � � � � 0, i = j and 1 ≤ |u|≤Z, 0, i ̸= j and 0 ≤ |u|≤Z, L, i = j and u = 0, (6) where 0 ≤ i, j ≤ K − 1 and Z is termed as ZCZ width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Definition 4: Let Z be a collection of N, (K, Z, L)-ZCZ sequence sets then Z = {Z1, Z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , ZN} is known as a multiple ZCZ sequence set with ZCCZ equal to Zc, if for 0 ≤ |u| < Zc, φ(zl i, zl′ j )(u) = 0, ∀1 ≤ l ̸= l′ ≤ N and 0 ≤ i, j ≤ K − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Definition 5 (Tang-Fan-Matsufuji Bound [28]): Let Z be any (K, Z, L)-ZCZ sequence set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Then, KZ ≤ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' If for any Z, KZ = L (or K(Z + 1) = L) then Z is called optimal (or near-optimal) ZCZ sequence set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' However, in case of binary ZCZ sequence set the bound is reduced to 2KZ ≤ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Generalised Boolean Function (GBF) [29] We define a complex valued sequence corresponding to a GBF, f : {0, 1}m −→ Zq of m variables as Ψ(f) = � ωf0, ωf1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , ωf2m−1� , (7) where fj = f(j0, j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , jm−1), ω = exp � 2π√−1/q � , and (j0, j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , jm−1) is the binary vector representation of j, where as in the remainder of this letter, q is an even integer not less than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Corresponding to a GBF f with m variables the sequence Ψ(f) is of length 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Definition 6: Let J = {j0, j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , jk−1} ⊂ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , n−1} and xJ = [xj0, xj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , xjk−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' For a constant e ∈ {0, 1}k, f|xJ=e is known as restriction of f over e and is obtained by substituting xjβ = eβ (β = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=', k − 1) in the function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Moreover, the sequence Ψ(f|xJ=e) is the same as sequence Ψ(f) of length 2m except for the positions ijβ ̸= eβ for each 0 ≤ β < k, at these positions Ψ(f|xJ=e) has the zero entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Quadratic Forms and Graphs [30] Let f be GBF of order r over m variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' If f|xJ=e is a quadratic GBF, then graph of f|xJ=e, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=', G(f|xJ=e) has vertex set V , where V = {x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , xm−1}\\{xj0, xj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , xjk−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' If there is a term qβ1β2xβ1xβ2 (0 ≤ β1 < β2 < m, xβ1, xβ2 ∈ V ) in the GBF f|xJ=e with qβ1β2 ̸= 0 (qβ1β2 ∈ Zq) then by connecting the vertices xβ1 and xβ2 by an edge, the graph G(f|xJ=e) can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' For k = 0, the graph of f|xJ=e is the same as that of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Generalized Reed-Muller Codes Definition 7: Let q ≥ 2 and 0 ≤ r ≤ m,, then a linear code over Zq generated by the Zq-valued sequences corresponding to the monomials of degree at most r in x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , xm−1 is said to be rth order generalised Reed-Muller (RM) code and denoted as RMq(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' The Existing Construction of Multiple CCCs Lemma 1 ( [31]): Let m, k, and s are integers with 0 ≤ s ≤ k ≤ m−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Define Js = {jk−s, jk−s+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , jk−1} = {m − s, m − s + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , m − 1}, J = {j0, j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , jk−1−s} ⊂ Zm−s, I = {i0, i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , im−k−1} = Zm−s\\J, x = � xj0, xj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , xjk−s−1 � , xs = � xjk−s, xjk−s+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , xjk−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Let π be a permutation on symbols {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , m − k − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Let f be a quadratic GBF over the m variables x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , xm−1, such that for e ∈ {0, 1}k−s, f|x=e = Q + m−k−1 � β=0 uβxiβ + s−1 � β=0 vβxjk−s+β + v, (8) where Q = q 2 m−k−2 � β=0 xiπ(β)xiπ(β+1), (9) uβ ∈ Zq ∀ 0 ≤ β ≤ m−k −1, vβ ∈ Zq ∀ 0 ≤ β ≤ s−1, and v ∈ Zq Let γ1 and γ2 be two end vertices of the path G(Q), t1 = �s−1 β=0 bk+1+β2β, t2 = �k β=0 bβ2β, where bβ ∈ {0, 1} for 0 ≤ β ≤ k +s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' For the natural order generated by (t1, t2), Define the set S(t1,t2) by � � �f + q 2 � � k−1 � β=0 dβxjβ + dxγ1 + k−1 � β=0 bβxjβ +bkxγ2 + k−1 � β=k−s dβbs+1+β � � : dβ, d ∈ {0, 1} � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' (10) Let St1 = � S(t1,t2) : 0 ≤ t2 ≤ 2k+1 − 1 � , 0 ≤ t1 ≤ 2s − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Then {St1 : 0 ≤ t1 ≤ 2s − 1} is a collection of 2s CCCs, and any two GCSs from different CCCs St1 and St′ 1 with 0 ≤ t1 ̸= t′ 1 ≤ 2s − 1 have a ZCCZ of width 2m−s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' For the fixed values of t1 and t2, S(t1,t2) is a GCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Let us denote, S(t1,t2) = � s(t1,t2) 0 s(t1,t2) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' s(t1,t2) 2k+1−1 �T , (11) 3 where s(t1,t2) ν (0 ≤ ν ≤ 2k+1 − 1) is νth row sequence of S(t1,t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Lemma 2 ( [13]): Let q = 2 and x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , xk, xk+1 be k+ 2 binary variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Also, let h be a Boolean function defined on x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , xk, xk+1 as follow h = k+1 � β=1 cβxβx0 + � 1≤µ<ν≤k dµνxµxν + k+1 � α=0 eαxα + e′, (14) where ck+1 = 1, cβ ∈ Z2 for 1 ≤ β ≤ k, dµν, eα, e′ ∈ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Let h denotes the binary vector corresponding to function h, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=', h = [h0, h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , h2k+2−1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' For 0 ≤ τ ≤ 2k+1 − 1, we have (−1)hτ +hτ+1 + (−1)hτ+2k+1+hτ+1+2k+1 = 0, (15) where the operation in the subscripts is done in modulo 2k+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' PROPOSED CONSTRUCTION In this section, we provide a GBF which generates the required multiple ZCZ sequence sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Theorem 1: Let x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , xm+k+1 are m + k + 2 binary variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Define a GBF f(x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , xm−1) on m variables same as in Lemma 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=', removing J = {j0, j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , jk−1−s} having k − s vertices from the graph of f results in s isolated vertices in Js and a path on m−k vertices in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Define another GBF h(xm, xm+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , xm+k+1) on k + 2 variables as h = k+1 � r=1 crxm+rxm + � 2≤µ<ν≤t dµνxm+µxm+ν + k+1 � β=1 eβxm+β + e′, (16) where ck+1 ̸= 0, cr ∈ Z2 for 1 ≤ r ≤ k, dµν, eβ ∈ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' For a fixed value of t1, define the set Zt1 = {Ψ(zt1 t2) : 0 ≤ t2 ≤ 2k+1 − 1} by � � �f + h + q 2 � � k−1 � β=0 xm+βxjβ + xm+kxγ1 + k−1 � β=0 bβxjβ +bkxγ2 + k−1 � β=k−s xm+βbs+1+β � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' (17) Then Z = � Zt1 : 0 ≤ t1 ≤ 2s − 1} is a collection of 2s (2k+1, 2m ,2m+k+2)-ZCZ sequence sets having ZCCZ equals to 2m−s − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Proof: Using (10), (11), (17) and taking natural order generated by t2, we get Zt1 = � Zt1 0 , Zt1 1 � , where � Zt1 0 , Zt1 1 � is horizontal concatenation of matrices Zt1 0 and Zt1 1 and these matrices are defined as, Zt1 0 = � ������ s(t1,0) 0 ωk0 s(t1,0) 1 ωk1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' s(t1,0) l−1 ωkl−1 s(t1,1) 0 ωk0 s(t1,1) 1 ωk1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' s(t1,1) l−1 ωkl−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' s(t1,l−1) 0 ωk0 s(t1,l−1) 1 ωk1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' s(t1,l−1) l−1 ωkl−1 � ������ , Zt1 1 = � ������ s(t1,0) 0 ωkl s(t1,0) 1 ωkl+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' s(t1,0) l−1 ωk2l−1 s(t1,1) 0 ωkl s(t1,1) 1 ωkl+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' s(t1,1) l−1 ωk2l−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' s(t1,l−1) 0 ωkl s(t1,l−1) 1 ωkl+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' s(t1,l−1) l−1 ωk2l−1 � ������ , where l = 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Now, we need to prove that Zt1 is a (2k+1, 2m , 2m+k+2)-ZCZ sequence set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' For 0 ≤ i, j ≤ 2k+1−1, periodic correlation of Ψ(zt1 i ) and Ψ(zt1 j ) at any time shift 0 ≤ τ ≤ 2m is given by (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Next, by (15), (12) and aperiodic sum property of CCCs, we get, φ(Ψ(zt1 i ), Ψ(zt1 j ))(τ) = 2 · 2k+1−1 � m=0 γ � ci m, cj m � (τ) (18) = � 2k+m+2, if τ = 0 and i = j, 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Which proves that Zt1 is a (2k+1, 2m, 2m+k+2)-ZCZ sequence sets ∀ 0 ≤ t1 ≤ 2s − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Now, let 0 ≤ t1 ̸= t′ 1 < 2s and 0 ≤ i, j ≤ 2k+1 − 1 then for 0 ≤ τ ≤ 2m−s − 1, the value of φ(Ψ(zt1 i ), Ψ(zt′ 1 j ))(τ) is given by (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Now, by (15), (13) and ZCCZ property of CCCs in Lemma 1, we get φ(Ψ(zt1 i ), Ψ(zt′ 1 j ))(τ) = 0, ∀ 0 ≤ τ ≤ 2m−s − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Remark 1: Theorem 1 constructed 2s ZCZ sequence sets with parameter (2k+1, 2m, 2m+k+2) having common ZCZ equals to 2m−s − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Since 2k+1 · 2m/2m+k+2 = 1/2 and Zc = 2m−s − 1 = (Z + 1)/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' φ(Ψ(zt1 i ), Ψ(zt1 j ))(τ) =2 · 2k+1−1 � m=0 γ � s(t1,i) m , s(t1,j) m � (τ) + [(−1)hl−1+hl + (−1)h2l−1+h0]γ∗ � s(t1,j) 0 , s(t1,i) 2l−1 � (L − τ) + 2l−2 � m=0 [(−1)hm+hm+1 + (−1)hm+2l+hm+1+2l]γ∗ � s(t1,j) m+1 , s(t1,i) m � (L − τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' (12) φ(Ψ(zt1 i ), Ψ(zt′ 1 j ))(τ) =2 · l−1 � m=0 γ � s(t1,i) m , s(t′ 1,j) m � (τ) + [(−1)hl−1+hl + (−1)h2l−1+h0]γ∗ � s(t′ 1,j) 0 , s(t1,i) 2l−1 � (L − τ) + l−2 � m=0 [(−1)hm+hm+1 + (−1)hm+l+hm+1+l]γ∗ � s(t′ 1,j) m+1 , s(t1,i) m � (L − τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' (13) 4 Remark 2: Since the set of isolated vertices in Theorem 1 contribute to multipleness of constructed multiple ZCZ sequence set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Hence, if we put s = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=', Js = φ in Theorem 1 then our construction reduces to construction presented in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Therefore, construction provided in [13] is a special case of the proposed construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Corollary 1: Collection of all the ZCZ sequences in Theo- rem 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=', {Ψ(zt1 t2) : 0 ≤ t2 ≤ 2k+1 − 1, 0 ≤ t1 ≤ 2s − 1} is a near-optimal (2k+s+1, 2m−s − 1, 2m+k+2)-ZCZ sequence set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Proof: Directly follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Remark 3: It is the first time in the literature that the direct construction of optimal multiple ZCZ sequence sets is provided such that their union is a near-optimal ZCZ sequence set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Which makes our construction advantageous over several constructions of A-ZCZ sequence sets which are presented in the literature [16], [18]–[23], [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' The detailed comparison of the proposed work is provided in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Remark 4: From equation (17), it can be seen that the proposed multiple ZCZ sequence sets are obtained from sec- ond order cosets of generalised RM code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Since, RM codes have efficient encoding, good error correction properties and important practical advantage of being easy to decode [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Hence, our proposed construction has advantage over any other non-GBF based construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' GRAPHICAL INTERPRETATION OF THE PROPOSED CONSTRUCTION This section interprets the proposed construction with graphical point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' 1 depicts a graphical repre- .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' 𝑖𝜋(0) 𝑖𝜋(𝑚−𝑘−1) 𝑖𝜋(1) 𝑖𝜋(2) 𝑗𝑘−1−𝑠 𝑗0 m-s m−1 m m+k-1-s I J J s Lower Layer Upper Layer .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' m+k m+k-1 m+k-s m+k+1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' 1: Graphical representation of (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' sentation of (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' The graph has a two-layered structure with a horizontal straight line which is separating the upper and bottom layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' The upper layer and lower layer correspond to graphs of Boolean functions f and h respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' These layers are interconnected through the set of edges {xj0xm, xj1xm+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , xjk−1−sxm+k−1−s, xm−sxm+k−s, xm−s+1xm+k−s+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' , xm−1xm+k−1}, and the vertex xm+k is connected to any of the end vertices of the path in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Interestingly, the ZCZ of each ZCZ sequence set is equals to the power of number of vertices in the upper layer of the graph and ZCCZ of ZCZ sequence sets equals to one less than the power of number of vertices in the upper layer of graph except isolated vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Example 1: Let m = 4, q = 2, s = 1, and k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Assume J = {0}, Js = {3}, I = {1, 2} and GBFs f = x0x1 + x0x2 + x0x3 + x1x2 + x1 + x2, h = x4x5 + x4x6 + x4x7 + x4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' (19) Generate two sequence sets Z0 and Z1 as Z0 = {Ψ(f+h+x0x4+x2x6+x3x5+b0·x0+b1·x3+b2·x1 + 0 · x5) : b0, b1, b2 ∈ Z2}, Z1 = {Ψ(f +h+x0x4+x2x6+x3x5+b0·x0+b1·x3+b2·x1 + 1 · x5) : b0, b1, b2 ∈ Z2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' (20) Then Z0 and Z1 are two optimal (8, 16, 256)-ZCZ sequence sets having inter-set ZCCZ equals to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Moreover, Z = Z0∪Z1 is also an optimal (16, 7, 256)-ZCZ sequence set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' 2, a graph corresponding to quadratic form, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=', f + h + x0x4 + x2x6 + x3x5 of Example 1 is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' 5 6 4 7 2 1 0 3 I J J s Upper Layer Lower Layer Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' 2: Graphical representation of f +h+x0x4+x2x6+x3x5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' CONCLUSION In this paper, we partially answered the open problem provided by Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' For the first time in the literature, we proposed a direct construction of multiple (2k+1, 2m, 2m+k+2)-ZCZ sequence sets having ZCCZ equals to (Z + 1)/N = 2m−s using GBF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' TABLE I: Comparison of the proposed construction with [19], [20], [22], [23], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Method Parameter1 Optimality2 ZCCZ No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' of sets Constraints [20, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' 1] Indirect (L, M − 1, LP) No 2M − 1 N N = ⌊ T M ⌋ > 1, L = KM, M > 1, K > 1 [20, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' 2] Indirect (T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' TL) No TL N N = ⌊ T M ⌋ > 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' L = KM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' M > 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' K > 1 [19] Indirect (M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' M − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' PM) Yes PM − 1 N N = ⌊ T M ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' N > 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' M > 1 [22] Indirect (L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' TLP) No TLP T gcd(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' P) = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' gcd(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' P) = 1(orL|PorP|L) [23] Indirect (2M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' 2TP) No 2TP T ⌊ P −2 Z ⌋ = M or ⌊ P −1 Z ⌋ = M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Z ≤ 2 [26] Indirect (N 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' N) Yes Z + 1 M N is order of DFT matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' N = M(Z + 1) This paper Direct (2k+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' 2m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' 2m+k+2) Yes 2m−s − 1 2s 0 ≤ s ≤ k ≤ m − 2 1 Parameter of each ZCZ sequence set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' 2 Optimality of each ZCZ sequence set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' 5 REFERENCES [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Fan, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Yuan, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Tu, “Z-complementary binary sequences,” IEEE Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dA0T4oBgHgl3EQfNP-W/content/2301.02144v1.pdf'} +page_content=' Lett.' 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