Research on Construction Methods of Low Correlation zone Complementary Sequence Sets
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摘要: 完备互补序列是一类具有理想相关函数性质的信号，在多址接入通信系统、雷达波形设计等领域具有广泛的应用。然而完备互补序列集合大小不超过其子序列数目。为扩展互补序列数目，该文研究了非周期低相关区互补序列集的构造方法，首先提出了两类有限域上的映射函数，进而得到两类参数渐近达到最优的低相关区互补序列集。该类低相关区互补序列集相比完备互补序列集具有更多的序列数目，在通信系统中可支持更多的用户。Abstract: Perfect complementary sequence is a kind of signal with ideal correlation function, which is widely used in multiple access communication system, radar waveform design and so on. However, the set size of perfect complementary sequences is at most equal to the number of its subsequences. In order to expand the number of complementary sequences, the construction methods of aperiodic low correlation zone complementary sequence set are studied in this paper. First, two kinds of mapping functions on finite fields are proposed, and then two kinds of low correlation zone complementary sequence set with asymptotically optimal parameters are obtained. The number of these kinds of low correlation zone complementary sequence set are more than the perfect complementary sequence set, and which could support more users in the communication system.
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表 2 例1中集合$ C $的部分互补序列
$ {C^{(0,0,0)}} $ $ {C^{(0,0,1)}} $ $ {C^{(0,1,0)}} $ $ {C^{(0,1,1)}} $ $ \begin{gathered} \omega _{56}^0\omega _{56}^0\omega _{56}^0\omega _{56}^0\omega _{56}^0\omega _{56}^0\omega _{56}^0 \\ \omega _{56}^7\omega _{56}^{14}\omega _{56}^{28}\omega _{56}^{21}\omega _{56}^{42}\omega _{56}^{49}\omega _{56}^{35} \\ \omega _{56}^{14}\omega _{56}^{28}\omega _{56}^0\omega _{56}^{42}\omega _{56}^{28}\omega _{56}^{42}\omega _{56}^{14} \\ \omega _{56}^{21}\omega _{56}^{42}\omega _{56}^{28}\omega _{56}^7\omega _{56}^{14}\omega _{56}^{35}\omega _{56}^{49} \\ \omega _{56}^{28}\omega _{56}^0\omega _{56}^0\omega _{56}^{28}\omega _{56}^0\omega _{56}^{28}\omega _{56}^{28} \\ \omega _{56}^{35}\omega _{56}^{14}\omega _{56}^{28}\omega _{56}^{49}\omega _{56}^{42}\omega _{56}^{21}\omega _{56}^7 \\ \omega _{56}^{42}\omega _{56}^{28}\omega _{56}^0\omega _{56}^{14}\omega _{56}^{28}\omega _{56}^{14}\omega _{56}^{42} \\ \omega _{56}^{49}\omega _{56}^{42}\omega _{56}^{28}\omega _{56}^{35}\omega _{56}^{14}\omega _{56}^7\omega _{56}^{21} \\ \end{gathered} $ $ \begin{gathered} \omega _{56}^0\omega _{56}^0\omega _{56}^0\omega _{56}^0\omega _{56}^0\omega _{56}^0\omega _{56}^0 \\ \omega _{56}^{21}\omega _{56}^{42}\omega _{56}^{49}\omega _{56}^{35}\omega _{56}^7\omega _{56}^{14}\omega _{56}^{28} \\ \omega _{56}^{42}\omega _{56}^{28}\omega _{56}^{42}\omega _{56}^{14}\omega _{56}^{14}\omega _{56}^{28}\omega _{56}^0 \\ \omega _{56}^7\omega _{56}^{14}\omega _{56}^{35}\omega _{56}^{49}\omega _{56}^{21}\omega _{56}^{42}\omega _{56}^{28} \\ \omega _{56}^{28}\omega _{56}^0\omega _{56}^{28}\omega _{56}^{28}\omega _{56}^{28}\omega _{56}^0\omega _{56}^0 \\ \omega _{56}^{49}\omega _{56}^{42}\omega _{56}^{21}\omega _{56}^7\omega _{56}^{35}\omega _{56}^{14}\omega _{56}^{28} \\ \omega _{56}^{14}\omega _{56}^{28}\omega _{56}^{14}\omega _{56}^{42}\omega _{56}^{42}\omega _{56}^{28}\omega _{56}^0 \\ \omega _{56}^{35}\omega _{56}^{14}\omega _{56}^7\omega _{56}^{21}\omega _{56}^{49}\omega _{56}^{42}\omega _{56}^{28} \\ \end{gathered} $ $ \begin{gathered} \omega _{56}^0\omega _{56}^8\omega _{56}^{16}\omega _{56}^{24}\omega _{56}^{32}\omega _{56}^{40}\omega _{56}^{48} \\ \omega _{56}^7\omega _{56}^{22}\omega _{56}^{44}\omega _{56}^{45}\omega _{56}^{18}\omega _{56}^{33}\omega _{56}^{27} \\ \omega _{56}^{14}\omega _{56}^{36}\omega _{56}^{16}\omega _{56}^{10}\omega _{56}^4\omega _{56}^{26}\omega _{56}^6 \\ \omega _{56}^{21}\omega _{56}^{50}\omega _{56}^{44}\omega _{56}^{31}\omega _{56}^{46}\omega _{56}^{19}\omega _{56}^{41} \\ \omega _{56}^{28}\omega _{56}^8\omega _{56}^{16}\omega _{56}^{52}\omega _{56}^{32}\omega _{56}^{12}\omega _{56}^{20} \\ \omega _{56}^{35}\omega _{56}^{22}\omega _{56}^{44}\omega _{56}^{17}\omega _{56}^{18}\omega _{56}^5\omega _{56}^{55} \\ \omega _{56}^{42}\omega _{56}^{36}\omega _{56}^{16}\omega _{56}^{38}\omega _{56}^4\omega _{56}^{54}\omega _{56}^{34} \\ \omega _{56}^{49}\omega _{56}^{50}\omega _{56}^{44}\omega _{56}^3\omega _{56}^{46}\omega _{56}^{47}\omega _{56}^{13} \\ \end{gathered} $ $ \begin{gathered} \omega _{56}^0\omega _{56}^8\omega _{56}^{16}\omega _{56}^{24}\omega _{56}^{32}\omega _{56}^{40}\omega _{56}^{48} \\ \omega _{56}^{21}\omega _{56}^{50}\omega _{56}^9\omega _{56}^3\omega _{56}^{39}\omega _{56}^{54}\omega _{56}^{20} \\ \omega _{56}^{42}\omega _{56}^{36}\omega _{56}^2\omega _{56}^{38}\omega _{56}^{46}\omega _{56}^{12}\omega _{56}^{48} \\ \omega _{56}^7\omega _{56}^{22}\omega _{56}^{51}\omega _{56}^{17}\omega _{56}^{53}\omega _{56}^{26}\omega _{56}^{20} \\ \omega _{56}^{28}\omega _{56}^8\omega _{56}^{44}\omega _{56}^{52}\omega _{56}^4\omega _{56}^{40}\omega _{56}^{48} \\ \omega _{56}^{49}\omega _{56}^{50}\omega _{56}^{37}\omega _{56}^{31}\omega _{56}^{11}\omega _{56}^{54}\omega _{56}^{20} \\ \omega _{56}^{14}\omega _{56}^{36}\omega _{56}^{30}\omega _{56}^{10}\omega _{56}^{18}\omega _{56}^{12}\omega _{56}^{48} \\ \omega _{56}^{35}\omega _{56}^{22}\omega _{56}^{23}\omega _{56}^{45}\omega _{56}^{25}\omega _{56}^{26}\omega _{56}^{20} \\ \end{gathered} $ $ {C^{(1,0,0)}} $ $ {C^{(1,0,1)}} $ $ {C^{(1,1,0)}} $ $ {C^{(1,1,1)}} $ $ \begin{gathered} \omega _{56}^0\omega _{56}^0\omega _{56}^0\omega _{56}^0\omega _{56}^0\omega _{56}^0\omega _{56}^0 \\ \omega _{56}^0\omega _{56}^{21}\omega _{56}^{35}\omega _{56}^{14}\omega _{56}^{49}\omega _{56}^{42}\omega _{56}^{28} \\ \omega _{56}^0\omega _{56}^{42}\omega _{56}^{14}\omega _{56}^{28}\omega _{56}^{42}\omega _{56}^{28}\omega _{56}^0 \\ \omega _{56}^0\omega _{56}^7\omega _{56}^{49}\omega _{56}^{42}\omega _{56}^{35}\omega _{56}^{14}\omega _{56}^{28} \\ \omega _{56}^0\omega _{56}^{28}\omega _{56}^{28}\omega _{56}^0\omega _{56}^{28}\omega _{56}^0\omega _{56}^0 \\ \omega _{56}^0\omega _{56}^{49}\omega _{56}^7\omega _{56}^{14}\omega _{56}^{21}\omega _{56}^{42}\omega _{56}^{28} \\ \omega _{56}^0\omega _{56}^{14}\omega _{56}^{42}\omega _{56}^{28}\omega _{56}^{14}\omega _{56}^{28}\omega _{56}^0 \\ \omega _{56}^0\omega _{56}^{35}\omega _{56}^{21}\omega _{56}^{42}\omega _{56}^7\omega _{56}^{14}\omega _{56}^{28} \\ \end{gathered} $ $\begin{gathered} \omega _{56}^0\omega _{56}^0\omega _{56}^0\omega _{56}^0\omega _{56}^0\omega _{56}^0\omega _{56}^0 \\ \omega _{56}^{14}\omega _{56}^{49}\omega _{56}^{42}\omega _{56}^{28}\omega _{56}^0\omega _{56}^{21}\omega _{56}^{35} \\ \omega _{56}^{28}\omega _{56}^{42}\omega _{56}^{28}\omega _{56}^0\omega _{56}^0\omega _{56}^{42}\omega _{56}^{14} \\ \omega _{56}^{42}\omega _{56}^{35}\omega _{56}^{14}\omega _{56}^{28}\omega _{56}^0\omega _{56}^7\omega _{56}^{49} \\ \omega _{56}^0\omega _{56}^{28}\omega _{56}^0\omega _{56}^0\omega _{56}^0\omega _{56}^{28}\omega _{56}^{28} \\ \omega _{56}^{14}\omega _{56}^{21}\omega _{56}^{42}\omega _{56}^{28}\omega _{56}^0\omega _{56}^{49}\omega _{56}^7 \\ \omega _{56}^{28}\omega _{56}^{14}\omega _{56}^{28}\omega _{56}^0\omega _{56}^0\omega _{56}^{14}\omega _{56}^{42} \\ \omega _{56}^{42}\omega _{56}^7\omega _{56}^{14}\omega _{56}^{28}\omega _{56}^0\omega _{56}^{35}\omega _{56}^{21} \\ \end{gathered} $ $\begin{gathered} \omega _{56}^0\omega _{56}^8\omega _{56}^{16}\omega _{56}^{24}\omega _{56}^{32}\omega _{56}^{40}\omega _{56}^{48} \\ \omega _{56}^0\omega _{56}^{29}\omega _{56}^{51}\omega _{56}^{38}\omega _{56}^{25}\omega _{56}^{26}\omega _{56}^{20} \\ \omega _{56}^0\omega _{56}^{50}\omega _{56}^{30}\omega _{56}^{52}\omega _{56}^{18}\omega _{56}^{12}\omega _{56}^{48} \\ \omega _{56}^0\omega _{56}^{15}\omega _{56}^9\omega _{56}^{10}\omega _{56}^{11}\omega _{56}^{54}\omega _{56}^{20} \\ \omega _{56}^0\omega _{56}^{36}\omega _{56}^{44}\omega _{56}^{24}\omega _{56}^4\omega _{56}^{40}\omega _{56}^{48} \\ \omega _{56}^0\omega _{56}^1\omega _{56}^{23}\omega _{56}^{38}\omega _{56}^{53}\omega _{56}^{26}\omega _{56}^{20} \\ \omega _{56}^0\omega _{56}^{22}\omega _{56}^2\omega _{56}^{52}\omega _{56}^{46}\omega _{56}^{12}\omega _{56}^{48} \\ \omega _{56}^0\omega _{56}^{43}\omega _{56}^{37}\omega _{56}^{10}\omega _{56}^{39}\omega _{56}^{54}\omega _{56}^{20} \\ \end{gathered} $ $ \begin{gathered} \omega _{56}^0\omega _{56}^8\omega _{56}^{16}\omega _{56}^{24}\omega _{56}^{32}\omega _{56}^{40}\omega _{56}^{48} \\ \omega _{56}^{14}\omega _{56}^1\omega _{56}^2\omega _{56}^{52}\omega _{56}^{32}\omega _{56}^5\omega _{56}^{27} \\ \omega _{56}^{28}\omega _{56}^{50}\omega _{56}^{44}\omega _{56}^{24}\omega _{56}^{32}\omega _{56}^{26}\omega _{56}^6 \\ \omega _{56}^{42}\omega _{56}^{43}\omega _{56}^{30}\omega _{56}^{52}\omega _{56}^{32}\omega _{56}^{47}\omega _{56}^{41} \\ \omega _{56}^0\omega _{56}^{36}\omega _{56}^{16}\omega _{56}^{24}\omega _{56}^{32}\omega _{56}^{12}\omega _{56}^{20} \\ \omega _{56}^{14}\omega _{56}^{29}\omega _{56}^2\omega _{56}^{52}\omega _{56}^{32}\omega _{56}^{33}\omega _{56}^{55} \\ \omega _{56}^{28}\omega _{56}^{22}\omega _{56}^{44}\omega _{56}^{24}\omega _{56}^{32}\omega _{56}^{54}\omega _{56}^{34} \\ \omega _{56}^{42}\omega _{56}^{15}\omega _{56}^{30}\omega _{56}^{52}\omega _{56}^{32}\omega _{56}^{19}\omega _{56}^{13} \\ \end{gathered} $ 
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表 1 渐近最优LCZ-CSS参数
$ p $,$ n $,$ q $ $ K $ $ M $ $ N $ $ Z $ $ {\rho _{{\text{LCZ}}}} $ $ p = 2,n = 3,q = 8 $ 112 8 7 3 1.3655 $ p = 3,n = 2,q = 9 $ 144 9 8 3 1.3312 $ p = 2,n = 4,q = 16 $ 1200 16 15 3 1.1434 $ p = 5,n = 2,q = 25 $ 4800 25 24 3 1.0871 $ p = 7,n = 2,q = 49 $ 15872 32 31 3 1.0426
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表 3 例2中集合$ C $的部分互补序列
$ {C^{(0,0,0)}} $ $ {C^{(0,0,1)}} $ $ {C^{(0,1,0)}} $ $ {C^{(0,1,1)}} $ $ {C^{(0,2,0)}} $ $ {C^{(0,2,1)}} $ $ {C^{(1,0,0)}} $ $ {C^{(1,0,1)}} $ $ {C^{(1,1,0)}} $ $ {C^{(1,1,1)}} $ $ \begin{gathered} \begin{array}{*{20}{l}} {132645} \\ {264513} \\ {326451} \\ {451326} \\ {513264} \\ {645132} \end{array} \\ 000000 \\ \end{gathered} $ $ \begin{array}{*{20}{l}} {645132} \\ {513264} \\ {451326} \\ {326451} \\ {264513} \\ {132645} \\ {000000} \end{array} $ $ \begin{array}{*{20}{l}} {326451} \\ {451326} \\ {513264} \\ {645132} \\ {000000} \\ {132645} \\ {264513} \end{array} $ $ \begin{array}{*{20}{l}} {451326} \\ {326451} \\ {264513} \\ {132645} \\ {000000} \\ {645132} \\ {513264} \end{array} $ $ \begin{gathered} \begin{array}{*{20}{l}} {264513} \\ {326451} \\ {451326} \\ {513264} \\ {645132} \\ {000000} \end{array} \\ 132645 \\ \end{gathered} $ $ \begin{array}{*{20}{l}} {513264} \\ {451326} \\ {326451} \\ {264513} \\ {132645} \\ {000000} \\ {645132} \end{array} $ $ \begin{array}{*{20}{l}} {132645} \\ {305624} \\ {541603} \\ {014652} \\ {250631} \\ {423610} \\ {666666} \end{array} $ $ \begin{array}{*{20}{l}} {645132} \\ {624305} \\ {603541} \\ {652014} \\ {631250} \\ {610423} \\ {666666} \end{array} $ $ \begin{gathered} \begin{array}{*{20}{l}} {326451} \\ {562430} \\ {035416} \\ {201465} \\ {444444} \\ {610423} \end{array} \\ 153402 \\ \end{gathered} $ $ \begin{array}{*{20}{l}} {451326} \\ {430562} \\ {416035} \\ {465201} \\ {444444} \\ {423610} \\ {402153} \end{array} $
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表 4 渐近最优非周期QCSS参数比较
文献 $ K $ $ M $ $ N $ $Z$ $ {\delta _{\max }} $ 字符集大小 约束条件 文献[10]定理4 $ N{q_1}{q_2} $，其中
$ N = {\text{lcm}}({q_1} - 1,{q_2} - 1) $$ {q_1}{q_2} $ $ {\text{lcm}}({q_1} - 1,{q_2} - 1) $ $ {q_1} - 1 $ $ {q_1}{q_2} $ $ {\text{lcm}}(N,{q_1}{q_2}) $ $ {q_1} $,$ {q_2} $为素数幂$ 3 \le {q_1} < {q_2} $ 文献[19] $ {2^n}({2^n} - 1)\left\lfloor {{{({2^n} - 1)} \mathord{\left/ {\vphantom {{({2^n} - 1)} Z}} \right. } Z}} \right\rfloor $ $ {2^n} $ $ {2^n} - 1 $ $ 0 < Z \le {2^n} - 1 $ $ {2^n} $ 2 $ n \ge 3 $ 文献[20] $N \times F(N)$ $N$ $N$ $N$ $N$ $N$ $N \ge 2$为任意整数 文献[21] ${p^{n + 1}}(p - 1)$ ${p^{n + 1}}$ ${p^m}$ ${p^m}$ ${p^m}$ $q$ $n \le m - 1$是任意非负整数，$p|q$ 本文定理1 $ q(q - 1)\left\lfloor {{{(q - 1)} \mathord{\left/ {\vphantom {{(q - 1)} Z}} \right. } Z}} \right\rfloor $ $ q $ $ q - 1 $ $0 < Z \le q - 1$ $ q $ $ q(q - 1) $ $ q \ge 3 $是素数幂 本文定理2 $ {q^2}\left\lfloor {{{(q - 1)} \mathord{\left/ {\vphantom {{(q - 1)} Z}} \right. } Z}} \right\rfloor $ $ q $ $ q - 1 $ $0 < Z \le q - 1$ $ q $ $ q $ 注：${\text{lcm}}$表示最大公约数，$F(N)$是$F(N) \times N$佛罗伦萨矩阵存在的最大行数，$p$是素数。
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