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 that of the perfect complementary sequence set, thereby supporting more users in the communication system.
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表 1 例1中集合$ {\boldsymbol{C}} $的部分互补序列
$ {{\boldsymbol{C}}^{(0,0,0)}} $ $ {{\boldsymbol{C}}^{(0,0,1)}} $ $ {{\boldsymbol{C}}^{(0,1,0)}} $ $ {{\boldsymbol{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} $ $ {{\boldsymbol{C}}^{(1,0,0)}} $ $ {{\boldsymbol{C}}^{(1,0,1)}} $ $ {{\boldsymbol{C}}^{(1,1,0)}} $ $ {{\boldsymbol{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 2 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$ {\boldsymbol{C}} $的部分互补序列
$ {{\boldsymbol{C}}^{(0,0,0)}} $ $ {{\boldsymbol{C}}^{(0,0,1)}} $ $ {{\boldsymbol{C}}^{(0,1,0)}} $ $ {{\boldsymbol{C}}^{(0,1,1)}} $ $ {{\boldsymbol{C}}^{(0,2,0)}} $ $ {{\boldsymbol{C}}^{(0,2,1)}} $ $ {{\boldsymbol{C}}^{(1,0,0)}} $ $ {{\boldsymbol{C}}^{(1,0,1)}} $ $ {{\boldsymbol{C}}^{(1,1,0)}} $ $ {{\boldsymbol{C}}^{(1,1,1)}} $ $ \begin{gathered} \begin{array}{*{20}{l}} {132645} \\ {264513} \\ {326451} \\ {451326} \\ {513264} \\ {645132} \\ 000000 \end{array} \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} \\ 132645 \end{array} \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}\\153402 \end{array} \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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