diff --git "a/7NE0T4oBgHgl3EQffQAA/content/tmp_files/load_file.txt" "b/7NE0T4oBgHgl3EQffQAA/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/7NE0T4oBgHgl3EQffQAA/content/tmp_files/load_file.txt" @@ -0,0 +1,660 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf,len=659 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='02400v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='IT] 6 Jan 2023 Springer Nature 2021 LATEX template A Direct Construction of Optimal 2D-ZCACS with Flexible Array Size and Large Set Size Gobinda Ghosh1, Sudhan Majhi2* and Shubhabrata Paul1 1Mathematics, IIT Patna, Bihta, Patna, 801103, Bihar, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 2*Electrical Communication Engineering, IISc Bangalore, CV Raman Rd, Bengaluru, 560012, Karnataka, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Corresponding author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' E-mail(s): smajhi@iisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='in;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Contributing authors: gobinda 1921ma06@iitp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='in;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' shubhabrata@iitp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='in;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Abstract In this paper, we propose a direct construction of optimal two- dimensional Z-complementary array code sets (2D-ZCACS) using multivariable functions (MVFs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' In contrast to earlier works, the proposed construction allows for a flexible array size and a large set size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Additionally, the proposed design can be transformed into a one-dimensional Z-complementary code set (1D-ZCCS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Many of the 1D-ZCCS described in the literature appeared to be special cases of this proposed construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' At last, we compare our work with the current state of the art and then draw our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Keywords: Two dimensional complete complementary codes (2D-CCC), multivariable function (MVF), two dimensional Z- complementary array code set (2D-ZCACS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 1 Introduction For an asynchronous two dimensional multi-carrier code-division multiple access (2D-MC-CDMA) system, the ideal 2D correlation properties of two dimensional complete complementary codes (2D-CCCs)[1] can be properly uti- lized to obtain interference-free performance [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Similar to one dimensional complete complementary code (1D-CCC)[3–5], one of the most significant 1 Springer Nature 2021 LATEX template 2 A Direct Construction of Optimal 2D-ZCACS drawbacks of 2D-CCC is that the set size is restricted [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Motivated by the scarcity of 2D-CCC with flexible set sizes, Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' proposed 2D Z- complementary array code sets (2D-ZCACSs) in [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' For a 2D − (K, Z1 × Z2)−ZCACSL1×L2 M , K, Z1×Z2, L1×L2 and M denote the set size, two dimen- sional zero-correlation zone (2D-ZCZ) width, array size and the number of constituent arrays, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' In [6, 7], authors obtained ternary 2D-ZCACSs by inserting some zeros into the existing binary 2D-ZCACSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' In 2021, Pai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' presented a new construction method of 2D binary Z-complementary array pairs (2D-ZCAP) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Recently, Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' in [9] proposed a construction of 2D-ZCACS by using Z-paraunitary (ZPU) matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' All these construc- tions of 2D-ZCACS depend heavily on initial sequences and matrices which increase hardware storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' For the first time in the literature, Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' in [10] proposed a direct construction of 2D-ZCACS based on MVF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' The array size of the proposed 2D-ZCACS is of the form L1 × L2, where L1 = 2m, L2 = 2pm1 1 pm2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' pmk k , m ≥ 1, mi ≥ 2 and the set size is of the form 2p2 1p2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' p2 k where pi is a prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Therefore the array size and the set size is restricted to some even numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Existing array and set size limitations through direct construction in the literature motivates us to search multivariable function (MVF) for more flex- ible array and set sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Our proposed construction provides 2D-ZCACS with parameter 2D − (R1R2M1M2, N1 × N2) − ZCACSR1N1×R2N2 M1M2 where M1 = �a i=1 pki i , M2 = �b j=1 qtj j , pi is any prime or 1, qj is prime, a, b, ki, tj ≥ 1, R1 and R2 are positive integer, such that R1 ≥ 1 and R2 ≥ 2, N1 = �a i=1 pmi i , N2 = �b j=1 qnj j , mi, nj ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' The set size in our proposed 2D-ZCACS construc- tion, R1R2M1M2, is more adaptable than the set size of 2D-ZCACS given in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Unlike [10], the proposed 2D-ZCACS can be reduced to 1D-ZCCS [11–18] also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' As a result, many existing optimal 1D-ZCCSs have become special cases of the proposed construction [16–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' The proposed construction also derived a new set of optimal 1D-ZCCS that had not previously been presented by direct method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Section 2 discusses construc- tion related definitions and lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Section 3 contains the construction of 2D-ZCACS and the comparison with the existing state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Finally, in Section 4, the conclusions are drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 2 Notations and definitions The following notations will be followed throughout this paper: ωn = exp � 2π√−1/n � , An = {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', n− 1} ⊂ Z, where n is a positive integer and Z is the ring of integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='1 Two Dimensional Array Definition 1 ([9]) Let A = � ag,i � and B = � bg,i � be complex-valued arrays of size l1 × l2 where 0 ≤ g < l1, 0 ≤ i < l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' The two dimensional aperiodic cross correlation Springer Nature 2021 LATEX template A Direct Construction of Optimal 2D-ZCACS 3 function (2D-ACCF) of arrays A and B at shift (τ1, τ2) is defined as C (A, B) (τ1, τ2) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 �l1−1−τ1 g=0 �l2−1−τ2 i=0 ag,ib∗ g+τ1,i+τ2, if 0 ≤ τ1 < l1, 0 ≤ τ2 < l2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' �l1−1−τ1 g=0 �l2−1+τ2 i=0 ag,i−τ2b∗ g+τ1,i, if 0 ≤ τ1 < l1, −l2 < τ2 < 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' �l1−1+τ1 g=0 �l2−1−τ2 i=0 ag−τ1,ib∗ g,i+τ2, if −l1 < τ1 < 0, 0 ≤ τ2 < l2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' �l1−1+τ1 g=0 �l2−1+τ2 i=0 ag−τ1,i−τ2b∗ g,i, if −l1 < τ1 < 0, −l2 < τ2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Here, (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' )∗ denotes the complex conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' If A = B, then C (A, B) (τ1, τ2) is called the two dimensional aperiodic auto correlation function (2D-AACF) of A and referred to as C (A) (τ1, τ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' When l1 = 1, the complex-valued arrays A and B are reduced to one dimensional complex-valued sequences A = (aj)l2−1 j=0 and B = (bj)l2−1 j=0 with the corresponding one dimensional aperiodic cross correlation function (1D- ACCF) given by C(A, B)(τ2) = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 �l2−1−τ2 i=0 aib∗ i+τ2, 0 ≤ τ2 < l2, �l2+τ2−1 i=0 ai−τ2b∗ i , −l2 < τ2 < 0, 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (1) Definition 2 [19],[9] For a set of s sets of arrays A = � Ak | k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , s − 1}, each set Ak = � Ak 0, Ak 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , Ak s−1 � is composed of s arrays of size is l1 × l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' The set A is said to be 2D-CCC with parameters (s, s, l1, l2) if the following holds C � Ak, Ak′� (τ1, τ2) = s−1 � i=0 C � Ak i , Ak′ i � (τ1, τ2) = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 sl1l2, (τ1, τ2) = (0, 0), k = k′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 0, (τ1, τ2) ̸= (0, 0), k = k′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 0, k ̸= k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (2) Definition 3 [10],[9] Let z1, z2, l1, l2 are positive integers and z1 ≤ l1, z2 ≤ l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Consider the sets of ˆs set of arrays A = � Ak | k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , ˆs − 1}, where each set Ak = � Ak 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , Ak s−1 � is composed of s arrays of size l1 × l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' The set A is said to Springer Nature 2021 LATEX template 4 A Direct Construction of Optimal 2D-ZCACS be 2D − (ˆs, z1 × z2) − ZCACSl1×l2 s if the following holds C � Ak, Ak′� (τ1, τ2) = s−1 � i=0 C � Ak i , Ak′ i � (τ1, τ2) = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 sl1l2, (τ1, τ2) = (0, 0), k = k′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 0, (τ1, τ2) ̸= (0, 0),|τ1| < z1,|τ2| < z2, k = k′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 0, |τ1| < z1,|τ2| < z2, k ̸= k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (3) When z1 = l1, z2 = l2, ˆs = s the 2D-ZCACS becomes 2D-CCC[19, 20] with parameter (s, l1, l2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' It should be noted that for l1 = 1, each array Ak i becomes l2-length sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Therefore, 2D-ZCACS can be reduced to a conventional 1D-(ˆs, z2) − ZCCSl2 s [21], [22],[23], where, ˆs, s, z2, l2 represents no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' of set, set size, ZCZ width and sequence length respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Lemma 1 [9] For a 2D − (ˆs, z1 × z2) − ZCACSl1×l2 s , the following inequality holds ˆsz1z2 ≤ s (l1 + z1 − 1) (l2 + z2 − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (4) We called 2D-ZCACS is optimal if the following equality holds ˆs = s � l1 z1 �� l2 z2 � , (5) where ⌊.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='⌋ denotes the floor function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='2 Multivariable Function Let a, b, mi, and nj be positive integers for 1 ≤ i ≤ a and 1 ≤ j ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Let pi be any prime or 1, and qj be a prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' A multivariable function (MVF) can be defined as f : Am1 p1 × Am2 p2 × · · · × Ama pa × An1 q1 × An2 q2 × · · · × Anb qb → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Let c, d ≥ 0 be integers such that 0 ≤ c < r and 0 ≤ d < s where r = pm1 1 pm2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' pma a and s = qn1 1 qn2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' qnb b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Then c and d can be written as c = c1 + c2pm1 1 + · · · + capm1 1 pm2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' pma−1 a−1 , d = d1 + d2qn1 1 + · · · + dbqn1 1 qn2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' qnb−1 b−1 , (6) where, 0 ≤ ci < pmi i and 0 ≤ dj < qnj j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Let Ci = (ci,1, ci,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , ci,mi) ∈ Ami pi , be the vector representation of ci with base pi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', ci = �mi k=1 ci,kpk−1 i and Dj = (dj,1, dj,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , dj,nj) ∈ Anj qj be the vector representation of dj with base qj, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', dj = �nj l=1 dj,lql−1 j where 0 ≤ ci,k < pi, and 0 ≤ dj,l < qj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' We define vectors associated with c and d as φ(c) = (C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , Ca) ∈ Am1 p1 × Am2 p2 × · · · × Ama pa , φ(d) = (D1, D2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , Db) ∈ An1 q1 × An2 q2 × · · · × Anb qb , Springer Nature 2021 LATEX template A Direct Construction of Optimal 2D-ZCACS 5 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' We also define an array associated with f as ψλ(f) = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed ωf0,0 λ ωf0,1 λ · · ωf0,r−1 λ ωf1,0 λ ωf1,1 λ · · ωf1,r−1 λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' ωfs−1,0 λ ωfs−1,1 λ · · ωfs−1,r−1 λ \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , (7) where fc,d = f � φ(c), φ(d) � and λ is a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Lemma 2 ([24]) Let t and t′ be two non-negative integers, where t ̸= t′, and p is a prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Then p−1 � j=0 ω(t−t′)j p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (8) Let us consider the set C as C = � Am1 p1 × Am2 p2 × · · · × Ama pa � × � An1 q1 × An2 q2 × · · · × Anb qb � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (9) Let 0 ≤ γ < pm1 1 pm2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' pma a and 0 ≤ µ < qn1 1 qn2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' qnb b be positive integers such that γ = γ1 + a � i=2 γi \uf8eb \uf8ed i−1 � i1=1 p mi1 i1 \uf8f6 \uf8f8 , µ = µ1 + b � j=2 µj \uf8eb \uf8ed j−1 � j1=1 q nj1 j1 \uf8f6 \uf8f8 , (10) where 0 ≤ γi < pmi i and 0 ≤ µj < qnj j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Let γi = (γi,1, γi,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , γi,mi) ∈ Ami pi be the vector representation of γi with base pi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', γi = �mi k=1 γi,kpk−1 i , where 0 ≤ γi,k < pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Similarly µj = (µj,1, µj,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , µj,nj) ∈ Anj qj be the vector representation of µj with base qj i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', µj = �nj l=1 µj,lql−1 j where 0 ≤ µj,l < qj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Let φ(γ) = (γ1, γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , γa) ∈ Am1 p1 ×Am2 p2 × · · · × Ama pa , (11) be the vector associated with γ and φ(µ) = (µ1, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , µb) ∈ An1 q1×An2 q2 × · · · × Anb qb , (12) be the vector associated with µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Let πi and σj be any permutations of the set {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', mi} and {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', nj}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Let us also define the MVF Springer Nature 2021 LATEX template 6 A Direct Construction of Optimal 2D-ZCACS f : C → Z, as f(φ(γ), φ(µ)) = f (γ1, γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , γa, µ1, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , µb) = a � i=1 λ pi mi−1 � e=1 γi,πi(e)γi,πi(e+1) + a � i=1 mi � e=1 di,eγi,e + b � j=1 λ qj nj−1 � o=1 µj,σj(o)µj,σj(o+1) + b � j=1 nj � o=1 cj,oµj,o, (13) where di,e, cj,o ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', λ − 1} and λ = l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , pa, q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , qb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Let us define the set Θ and T as Θ = {θ : θ = (r1, r2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , ra, s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , sb)}, T = {t : t = (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , xa, y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , yb)}, where 0 ≤ ri, xi < pki i and 0 ≤ sj, yj < qrj j and ki, rj are positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Now, we define a function aθ t: C →Z, as aθ t � φ(γ), φ(µ) � = aθ t (γ1, γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , γa, µ1, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , µb) =f � φ(γ), φ(µ) � + a � i=1 λ pi γi,πi(1)ri + b � j=1 λ qj µj,σj(1)sj + a � i=1 λ pi γi,πi(mi)xi + b � j=1 λ qj µj,σj(nj)yj + dθ, (14) where 0 ≤ dθ < λ, γi,πi(1), γi,πi(mi) denote πi(1)−th and πi(mi)−th element of γi respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Similarly, µj,σj(1), µj,σj(nj) denote σj(1)−th and σj(nj)−th element of µj respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' For simplicity, we denote aθ t � φ(γ), φ(µ) � by (aθ t)γ,µ and f � φ(γ), φ(µ) � by fγ,µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Lemma 3 ([20]) We define the ordered set of arrays At = {ψλ � aθ t � : θ ∈ Θ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Then the set {At : t ∈ T } forms a 2D-CCC with parameter (α, α, m, n), where, α = �a i=1 pki i �b j=1 qrj j , m = �a i=1 pmi i , n = �b j=1 qnj j and ki, mi, nj, rj are non-negative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Springer Nature 2021 LATEX template A Direct Construction of Optimal 2D-ZCACS 7 3 Proposed construction of 2D-ZCACS Let a′, b′ be positive integers for 1 ≤ i′ ≤ a′ and 1 ≤ j′ ≤ b′, p′ i′ be any prime or 1, and q′ j′ be prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Let γ′, µ′ are positive integers such that 0 ≤ γ′ < ��a i=1 pmi i � ��a′ i′=1 p′ i′ � and 0 ≤ µ′ < ��b j=1 qnj j � ��b′ j′=1 q′ j′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Then γ′, µ′ can be written as γ′ =γ1+ a � i=2 γi \uf8eb \uf8ed i−1 � i1=1 p mi1 i1 \uf8f6 \uf8f8+ \uf8eb \uf8ec \uf8edγ′ 1 + a′ � i′=2 γ′ i′ \uf8eb \uf8ed i′−1 � i1=1 p′ i1 \uf8f6 \uf8f8 \uf8f6 \uf8f7 \uf8f8 m, µ′ =µ1+ b � j=2 µj \uf8eb \uf8ed j−1 � j1=1 q nj1 j1 \uf8f6 \uf8f8+ \uf8eb \uf8ec \uf8edµ′ 1 + b′ � j′=2 µ′ j′ \uf8eb \uf8ed j′−1 � j1=1 q′ j1 \uf8f6 \uf8f8 \uf8f6 \uf8f7 \uf8f8 n, (15) where m = �a i=1 pmi i , n = �b j=1 qnj j , 0 ≤ γi < pmi i , 0 ≤ µj < qnj j , 0 ≤ γ′ i′ < p′ i′ and 0 ≤ µ′ j′ < q′ j′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' We denote the vectors associated with γ′ and µ′ are φ(γ′) = � γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , γa, γ′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , γ′ a � ∈ Am1 p1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' × Ama pa × Ap′ 1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' × Ap′ a′ , φ(µ′) = � µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , µb, µ′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , µ′ b � ∈ An1 q1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' × Anb qb × Aq′ 1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' × Aq′ b′ , (16) respectively, where γi ∈ Ami pi , µj ∈ Anj qj are the vectors associated with γi and µj respectively i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', γi = (γi,1, γi,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , γi,mi) ∈ Ami pi , µj = (µj,1, µj,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , µj,nj) ∈ Anj qj , γi = �mi k=1 γi,kpk−1 i , µj = �nj l=1 µi,lql−1 j , 0 ≤ γi,k < pi and 0 ≤ µj,l < qj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Let us consider the set D as D = Am1 p1 ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='×Ama pa ×Ap′ 1 ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='×Ap′ a′ ×An1 q1 ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='×Anb qb ×Aq′ 1 ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='×Aq′ b′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (17) Let f be the function as defined (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' We define the MVF M c,d : D → Z as M c,d � φ(γ′), φ(µ′) � = M c,d � γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , γa, γ′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , γ′ a′, µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , µb, µ′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , µ′ b′ � = δ λf (γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , γa, µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , µb)+ a′ � i′=1 ci′ δ p′ i′ γ′ i′ + b′ � j′=1 dj′ δ q′ j′ µ′ j′, (18) where 0 ≤ ci′ < p′ i′, 0 ≤ dj′ < q′ j′, c = (c1, c2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , ca′) and d = (d1, d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , db′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' For simplicity, now on-wards we denote M c,d(γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , γa, γ′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , γ′ a′, µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , µb, µ′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , µ′ b′) by M c,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Consider the set Θ and T as Θ = {θ : θ = (r1, r2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , ra, s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , sb)}, T = {t : t = (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , xa, y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , yb)}, Springer Nature 2021 LATEX template 8 A Direct Construction of Optimal 2D-ZCACS where 0 ≤ ri, xi < pki i and 0 ≤ sj, yj < qrj j and ki, rj are positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Let us define MVF, bθ,c,d t : D → Z, as bθ,c,d t =M c,d + a � i=1 δ pi γi,πi(1)ri + b � j=1 δ qj µj,σj(1)sj + a � i=1 δ pi γi,πi(mi)xi + b � j=1 δ qj µj,σj(nj)yj + δ λdθ, (19) where 0 ≤ dθ < λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' By (14), (18) and (19) we have bθ,c,d t = δ λaθ t + a′ � i′=1 ci′ δ p′ i′ γ′ i′ + b′ � j′=1 dj′ δ q′ j′ µ′ j′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (20) We define the ordered set of arrays as Ωc,d t = {ψδ(bθ,c,d t ) : θ ∈ Θ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (21) where δ = l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='m(λ, p′ 1, p′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , p′ a′, q′ 1, q′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , q′ b′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Theorem 1 Let m = �a i=1 pmi i , n = �b j=1 qnj j , c = (c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , ca′), d = (d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , db′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Then the set S = {Ωc,d t : t ∈ T, 0 ≤ ci′ < p′ i′, 0 ≤ dj′ < q′ j′} forms a 2D − (α1, z1 × z2) − ZCACSl1×l2 α , where, α1 = ��a′ i′=1 p′ i′ � ��b′ j′=1 q′ j′ � α, l1 = m ��a′ i′=1 p′ i′ � , l2 = n ��b′ j′=1 q′ j′ � , z1 = m ,z2 = n, α = (�a i=1 pki i )(�b j=1 qrj j ), ki, rj, mi, nj ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Proof Let ˆγ, ˆµ are positive integers such that 0 ≤ ˆγ < l1 and 0 ≤ ˆµ < l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Then ˆγ, ˆµ can be written as ˆγ = γ1+ a � i=2 γi \uf8eb \uf8ed i−1 � i1=1 p mi1 i1 \uf8f6 \uf8f8+ \uf8eb \uf8ec \uf8edγ′ 1 + a′ � i′=2 γ′ i′ \uf8eb \uf8ed i′−1 � i1=1 p′ i1 \uf8f6 \uf8f8 \uf8f6 \uf8f7 \uf8f8 m, ˆµ = µ1+ b � j=2 µj \uf8eb \uf8ed j−1 � j1=1 qnj1 j1 \uf8f6 \uf8f8+ \uf8eb \uf8ec \uf8ec \uf8edµ′ 1 + b′ � j′=2 µ′ j′ \uf8eb \uf8ec \uf8ed j′−1 � j1=1 q′ j1 \uf8f6 \uf8f7 \uf8f8 \uf8f6 \uf8f7 \uf8f7 \uf8f8 n, where 0 ≤ γi < pmi i , 0 ≤ µj < qnj j , 0 ≤ γ′ i′ < p′ i′ and 0 ≤ µ′ j′ < q′ j′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' The proof will be split into following cases Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (τ1 = 0, τ2 = 0) Springer Nature 2021 LATEX template A Direct Construction of Optimal 2D-ZCACS 9 The ACCF between Ωc,d t and Ωc′,d′ t′ at τ1 = 0 and τ2 = 0 can be expressed as C(Ωc,d t , Ωc′,d′ t′ )(0, 0) = � θ∈Θ C(ψδ((bθ,c,d t )), ψδ((bθ,c′,d′ t′ )))(0, 0) = � θ∈Θ l1−1 � ˆγ=0 l2−1 � ˆµ=0 ω (bθ,c,d t )ˆγ,ˆ µ−(bθ,c′,d′ t′ )ˆγ,ˆ µ δ = � θ∈Θ m−1 � γ=0 n−1 � µ=0 p′ 1−1 � γ′ 1=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' p′ a′ −1 � γ′ a′=0 q′ 1−1 � µ1=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' q′ b′ −1 � µ′ b′ =0 ωD δ , (22) where D = δ λ � (aθ t )γ,µ − (aθ t′)γ,µ � +�a′ i′=1 δ p′ i′ (ci′ −c′ i′)γi′ +�b′ j′=1 δ q′ j′ (dj′ −d′ j′)µj′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' After splitting (22), we get C(Ωc,d t , Ωc′,d′ t′ )(0, 0) = \uf8eb \uf8ed� θ∈Θ m−1 � γ=0 n−1 � µ=0 ω δ λ � (aθ t )γ,µ−(aθ t′ )γ,µ � δ \uf8f6 \uf8f8 EF = \uf8eb \uf8ed� θ∈Θ m−1 � γ=0 n−1 � µ=0 ω � (aθ t )γ,µ−(aθ t′ )γ,µ � λ \uf8f6 \uf8f8 EF = C(At, At′ )(0, 0)EF, (23) where E = a′ � i′=1 \uf8eb \uf8ec \uf8ed p′ i′ −1 � γ′ i′ =0 ω (ci′−c′ i′)γ′ i′ p′ i′ \uf8f6 \uf8f7 \uf8f8 , F = b′ � j′=1 \uf8eb \uf8ec \uf8ec \uf8ed q′ j′ −1 � µ′ j′ =0 ω (dj′ −d′ j′ )µ′ j′ q′ j′ \uf8f6 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (24) Subcase (i): (t ̸= t′) By lemma 2 we know, the set {At : t ∈ T } forms a 2D-CCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Hence By lemma 2, we have C(At, At′ )(0, 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (25) Hence by (23) and (25) we have C(Ωc,d t , Ωc′,d′ t′ )(0, 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (26) Subcase (ii): (t = t′) By lemma 2, we know C(At, At′ )(0, 0) = \uf8eb \uf8ed a � i=1 pmi+ki i \uf8f6 \uf8f8 \uf8eb \uf8ed b � j=1 qnj+rj j \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (27) Springer Nature 2021 LATEX template 10 A Direct Construction of Optimal 2D-ZCACS Let M = ��a i=1 pmi+ki i � ��b j=1 qnj+rj j � hence by Lemma 2, (23), (24), (27), we have the following C(Ωc,d t , Ωc′,d′ t )(0, 0) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 M ��a′ i′=1 p′ i′ � ��b′ j′=1 q′ j′ � c = c′, d = d′ 0, c ̸= c′, d = d′ 0, c = c′, d ̸= d′ 0, c ̸= c′, d ̸= d′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (28) Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (0 < τ1 < �a i=1 pmi i , 0 < τ2 < �b j=1 qnj j ) Let σ, ρ are positive integers such that 0 ≤ σ < m′ and 0 ≤ ρ < n′ where m′ = �a′ i′=1 p′ i′, n′ = �b′ j′=1 q′ j′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Then σ and ρ can be written as σ = σ1 + σ2p′ 1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' + σa′ \uf8eb \uf8ed a′−1 � i′=1 p′ i′ \uf8f6 \uf8f8 , ρ = ρ1 + ρ2q′ 1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' + ρb′ \uf8eb \uf8ed b′−1 � j′=1 q′ j′ \uf8f6 \uf8f8 , (29) respectively where 0 ≤ σi′ < p′ i′ and 0 ≤ ρj′ < q′ j′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' We define vectors associated with σ and ρ to be φ(σ) = (σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , σa′) ∈ Ap′ 1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' × Ap′ a′ , φ(ρ) = (ρ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , ρb′) ∈ Aq′ 1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' × Aq′ b′ , (30) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' The ACCF between Ωc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='d t and Ωc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='d′ t′ for 0 < τ1 < �a i=1 pmi i and 0 < τ2 < �b j=1 qnj j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' can be derived as C(Ωc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='d t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Ωc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='d′ t′ )(τ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' τ2) =C(At,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' At′ )(τ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' τ2)DE+C(At,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' At′ )(τ1− a � i=1 pmi i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' τ2)D′E+ C(At,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' At′ )(τ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' τ2 − b � j=1 qnj j )DE′ + C(At,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' At′ )(τ1 − a � i=1 pmi i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' τ2 − b � j=1 qnj j )D′E′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (31) where D = m′−1 � σ=0 \uf8eb \uf8ed a′ � i′=1 ω (ci′−c′ i′ )(σi′ ) p′ i′ \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (32) E = n′−1 � ρ=0 \uf8eb \uf8ed b′ � j′=1 ω (dj′ −d′ j′)(ρj′ ) q′ j′ \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (33) D′ = m′−2 � σ=0 \uf8eb \uf8ed a′ � i′=1 ω(ci′(σi′ )−c′ i′ (σ+1)i′) p′ i′ \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (34) E′ = n′−2 � ρ=0 \uf8eb \uf8ed b′ � j′=1 ω � dj′ (ρj′ )−d′ j′(ρ+1)j′ � q′ j′ \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (35) Springer Nature 2021 LATEX template A Direct Construction of Optimal 2D-ZCACS 11 and (σ + 1)i′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (ρ + 1)j′ denotes the i′-th and j′-th components of φ (σ + 1) and φ (ρ + 1) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' By Lemma 2, for 0 < τ1 < �a i=1 pmi i and 0 < τ2 < �b j=1 qnj j , we have C(At, At′ )(τ1, τ2) = 0, (36) C(At, At′ )(τ1− a � i=1 pmi i , τ2) = 0, (37) C(At, At′ )(τ1, τ2 − b � j=1 qnj j ) = 0, (38) C(At, At′ )(τ1 − a � i=1 pmi i , τ2 − b � j=1 qnj j ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (39) By (31), (36), (37), (38), (39) we have C(Ωc,d t , Ωc ′ ,d ′ t′ )(τ1, τ2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (40) Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (0 < τ1 < �a i=1 pmi i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' − �b j=1 qnj j < τ2 < 0) The ACCF between Ωc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='d t and Ωc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='d′ t′ for 0 < τ1 < �a i=1 pmi i and − �b j=1 qnj j < τ2 < 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' can be derived as C(Ωc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='d t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Ωc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='d′ t′ )(τ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' τ2) =C(At,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' At′ )(τ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' τ2)DE+C(At,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' At′ )(τ1 − a � i=1 pmi i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' τ2)D′E + C(At,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' At′ )(τ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' b � j=1 qnj j + τ2)DE′′ + C(At,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' At′ )(τ1 − a � i=1 pmi i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' b � j=1 qnj j + τ2)D′E′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (41) where E′′ = n′−2 � ρ=0 \uf8eb \uf8ed b′ � j′=1 ω � dj′ (ρ+1)j′ −d′ j′ (ρj′ ) � q′ j′ \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (42) By Lemma 2, for 0 < τ1 < �a i=1 pmi i and − �b j=1 qnj j < τ2 < 0, we have C(At, At′ )(τ1, b � j=1 qnj j + τ2) = 0, (43) C(At, At′ )(τ1 − a � i=1 pmi i , b � j=1 qnj j + τ2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (44) By (41) , (43) and (44) we have C(Ωc,d t , Ωc′,d′ t′ )(τ1, τ2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (45) Case 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (0 < τ1 < �a i=1 pmi i , τ2 = 0) Springer Nature 2021 LATEX template 12 A Direct Construction of Optimal 2D-ZCACS The ACCF between Ωc,d t and Ωc′,d′ t′ for 0 < τ1 < �a i=1 pmi i and τ2 = 0 , can be derived as C(Ωc,d t , Ωc′,d′ t′ )(τ1, 0) =C(At, At′ )(τ1, 0)DE+ C(At, At′ )(τ1 − a � i=1 pmi i , 0)D′E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (46) By Lemma 2, for 0 < τ1 < �a i=1 pmi i , we have C(At, At′ )(τ1, 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' C(At, At′ )(τ1 − a � i=1 pmi i , 0) = 0, (47) by (46) and (47) we have C(Ωc,d t , Ωc′,d′ t′ )(τ1, 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (48) Case 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (τ1 = 0, 0 < τ2 < �b j=1 qnj j ) The ACCF between Ωc,d t and Ωc′,d′ t′ for τ1 = 0 and 0 < τ2 < �b j=1 qnj j , can be derived as C(Ωc,d t , Ωc′,d′ t′ )(0, τ2) =C(At, At′ )(0, τ2)DE+ C(At, At′ )(0, τ2 − b � j=1 qnj j )DE′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (49) By Lemma 2, for 0 < τ2 < �b j=1 qnj j , we have C(At, At′ )(0, τ2) = 0, C(At, At′ )(0, τ2 − b � j=1 qnj j ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (50) By (49) and (50) we have C(Ωc,d t , Ωc′,d′ t′ )(0, τ2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (51) Case 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (τ1 = 0, − �b j=1 qnj j < τ2 < 0) Similarly the ACCF between Ωc,d t and Ωc′,d′ t′ for τ1 = 0 and − �b j=1 qnj j < τ2 < 0 is C(Ωc,d t , Ωc′,d′ t′ )(0, τ2) =C(At, At′ )(0, τ2)DE+ C(At, At′ )(0, τ2 + b � j=1 qnj j )DE′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (52) By Lemma 2, for − �b j=1 qnj j < τ2 < 0, we have C(At, At′ )(0, τ2 + b � j=1 qnj j ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (53) Hence by (50), (52) and (53) we have C(Ωc,d t , Ωc′,d′ t′ )(0, τ2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (54) Springer Nature 2021 LATEX template A Direct Construction of Optimal 2D-ZCACS 13 Combining all the cases we have C(Ωc,d t , Ωc′,d′ t′ )(τ1, τ2) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 M ��a′ i′=1 p′ i′ � ��b′ j′=1 q′ j′ � , (c, d, t) = (c′, d′, t′) (τ1, τ2) = (0, 0), 0, (c, d, t) ̸= (c′, d′, t′) (τ1, τ2) = (0, 0), 0, 0 ≤ τ1 < �a i=1 pmi i , (τ1, τ2) ̸= (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (55) Similarly it can be shown C(Ωc,d t , Ωc′,d′ t′ )(τ1, τ2) = 0, − a � i=1 pmi i < τ1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (56) Hence from (55), (56) we derive our conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' □ Example 1 Suppose that a = 1, b = 1, a′ = 1, b′ = 1, p1 = 2, m1 = 2, k1 = 1, q1 = 3, n1 = 2, r1 = 1, p′ 1 = 3, q′ 1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Let δ = 6, λ = 6, γ1 = (γ11, γ12) ∈ A2 2 = {0, 1}2 be the vector associated with γ1 where 0 ≤ γ1 ≤ 3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', γ1 = γ11 + 2γ12 and µ1 = (µ11, µ12) ∈ A2 3 = {0, 1, 2}2 be the vector associated with µ1 where 0 ≤ µ1 ≤ 8, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', µ1 = µ11+3µ12 and 0 ≤ γ′ 1 ≤ 2, 0 ≤ µ′ 1 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' We define the MVF f : A2 2×A2 3 → Z as f (γ1, µ1)=3γ1,2γ1,1+γ1,1+2γ1,2+2µ1,2µ1,1+2µ1,1+µ1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Consider the MVF, Mc,d : A2 2 × A3 × A2 3 × A2 → Z as Mc,d � γ1, γ′ 1, µ1, µ′ 1 � = f(γ1, µ1) + 2c1γ′ 1 + 3d1µ′ 1 = 3γ1,2γ1,1 + γ1,1 + 2γ1,2 + 2µ1,2µ1,1 + 2µ1,1 + µ1,2 + 2c1γ′ 1 + 3d1µ′ 1, (57) where 0 ≤ c1 < p′ 1 = 2, 0 ≤ d1 < q′ 1 = 3, c = c1 ∈ {0, 1}, and d = d1 ∈ {0, 1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' We have Θ = {θ : θ = (r1, s1) : 0 ≤ r1 ≤ 1, 0 ≤ s1 ≤ 2}, T = {t : t = (x1, y1) : 0 ≤ x1 ≤ 1, 0 ≤ y1 ≤ 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (58) Let dθ = 0, now from (19) we have bθ,c,d t = Mc,d + 3γ1,2r1 + 2µ1,2s1 + 3γ1,1x1 + 2µ1,2y1, (59) and Ωc,d t = � ψ6(bθ,c,d t ) : θ = (r1, s1) ∈ {0, 1} × {0, 1, 2} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (60) Therefore, the set S = {Ωc,d t : t ∈ T, 0 ≤ c1 ≤ 1, 0 ≤ d1 ≤ 2}, (61) forms an optimal 2D − (36, 4 × 9) − ZCACS12×18 6 over Z6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 14 A Direct Construction of Optimal 2D-ZCACS Table 1 Comparison with Previous Works Source No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' of set Array Size Condition Based on [7] K = K′r L′ 1×(L′ 2 + r + 1) r ≥ 0 2D − ZCACS of set size K′ and array size L′ 1×L′ 2 [8] 1 2m × 2nL m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' n ≥ 0 ZCP of length L [9] K K × K K divides set size BH matrices [10] 2 �ki i=1 p2 i 2m × �ki i=1 pmi i ki,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' mi ≥ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' pi’s are prime MVF Thm 2 rsα rm × sn α = (�a i=1 pki i )(�b j=1 q rj j ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' m=�a i=1pmi i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' n=�b j=1q nj j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' α ≥ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' pi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' qjareprimes MVF Remark 1 In Theorem 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' if we take a = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' p1 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' a′ = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' p′ 1 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' b = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' q1 = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' b′ = l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' r1 ≥ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' we have optimal 1D-ZCCS with parameter (�l i=1 q′ i2r1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 2n1) − ZCCS �l i=1 q′ i2n1 2r1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' which is exactly the same result as in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Also if we take l = 1, then we have optimal 1D-ZCCS of the form (q′ 12r1, 2n1) − ZCCSq′ 12n1 2r1 , which is exactly the same result in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Therefore the optimal 1D-ZCCS given by [17, 18] appears as a special case of the proposed construction Remark 2 In Theorem 1, if a = 1, p1 = 1, a′ = 1, p′ 1 = 1, b = 1, q1 = 2, b′ = l, r1 = 1, we have 1d-ZCCS with parameter (2 �l i=1 q′ i, 2n1) − ZCCS �l i=1 q′ i2n1 2 , which is just a collection of 2 �l i=1 q′ i ZCPs with sequence length �l i=1 q′ i2n1 and ZCZ width 2n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Hence our work produces collections of ZCPs[15] as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Remark 3 In Theorem 1, if we take a = 1, p1 = 1, a′ = 1, p′ 1 = 1, b = 1, q1 = 2, b′ = r, q′ 1 = q′ 2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' = q′r = 2, n1 = m − r and r1 = s + 1 then we have 1D-ZCCS with parameter (2s+r+1, 2m−r)−ZCCS2m 2s+1, which is exactly the same result in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Hence, the ZCCS in [16] appears as a special case of our proposed construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Remark 4 The 2D-ZCACS given by the proposed construction satisfies the equality given in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Therefore the 2D-ZCACS obtained by the proposed construction is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Remark 5 If we take a = 1, a′ = 1, p1 = 1 and p′ 1 = 1, in Theorem 1, we have optimal 1D-ZCCS with parameter ���b′ j′=1 q′ j′ � �b j=1 qrj j , n � − ZCCS n ��b′ j′=1 q′ j′ � �b j=1 q rj j where, n = �b j=1 qnj j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Hence, we have optimal 1D-ZCCS of length nm where, n, m > 1 and m = �b′ j′=1 q′ j′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Therefore our construction produces optimal 1D-ZCCS with a new length which is not present in the literature by direct method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Springer Nature 2021 LATEX template A Direct Construction of Optimal 2D-ZCACS 15 Remark 6 The set size of our proposed 2D-ZCACS is ��a′ i′=1 p′ i′ � ��b′ j′=1 q′ j′ � �a i=1 pki i �b j=1 qrj j where, ki, tj ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' If we take a = 1, p1 = 1, a′ = 1, p′ 1 = 1, r1 = r2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' = rb = 2, b′ = 1, and q′ 1 = 2 then we have set size 2 �b j=1 q2 j which is the set size of the 2D-ZCACS in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Therefore, we have flexible number of set sizes compared to [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='1 Comparison with Previous Works Table I compares the proposed work with indirect constructions from [7–9] and direct construction from [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' The constructions in [7–9] heavily rely on initial sequences, increasing hardware storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' The construction in [10] is direct, but set size and array sizes are limited to some even numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Our construction doesn’t require initial matrices or sequences and produces flexible parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 4 Conclusion In this paper, 2D-ZCACSs are designed by using MVF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' The proposed design does not depend on initial sequences or matrices, so it is direct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Our proposed design produces flexible array size and set size compared to existing works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Also, our proposed construction can be reduced to 1D-ZCCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' As a result, many 1D-ZCCSs become special cases of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Finally, we compare our work to the existing state-of-the-art and show that it’s more versatile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' References [1] Farkas, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', 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