低（奇）偶周期相关序列的研究
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摘要
低(奇)偶周期相关序列在密码、码分多址通信系统、正交频分复用通信系统、编码、雷达、声纳等领域有着重要的应用。在许多通信系统中,序列性能的好坏直接影响着通信系统的性能优劣和系统容量的大小。本论文主要研究了具有低(奇)偶周期相关性质的序列,特别针对跳频序列的理论界及其最优构造、(几乎)完备和奇完备序列的构造、低/零奇相关区序列的构造、格雷和QAM格雷序列的完备和奇完备循环共轭性质研究、形似于格雷序列的周期互补对和奇周期互补对的构造、二元签名序列的理论界及其最优构造等五个方面进行了深入研究。
首先,根据Song等人ODing等人的工作,本文进一步研究了跳频序列集和循环码之间存在密切的联系,讨论了已知的理论界之间的关系。发现了从Plotkin编码理论界得到的Peng-Fan界Plotkin跳频序列理论界的之间的联系、从Singleton编码理论界出发得到的两个跳频序列理论界之间的联系以及两个Peng-Fan界之间的关系。除此之外,基于一些的循环码,包括Reed-Solomon码以及其截短码、多项式函数定义的码、以及两类MDS码及其截短的MDS码,设计了达到新的最大汉明相关的Singleton界的最优跳频序列集。
其次研究了(几乎)完备和奇完备序列的构造。利用差平衡函数,定义了新的移位序列,推广了Krengel的几乎完备和奇完备三元序列的构造方法。选取平衡的几乎完备序列,构造了(几乎)完备和奇完备的三元序列、高斯整数序列以及QAM+序列,还设计了具有几乎完备性质的16-QAM序列。论文将利用素数p的2阶和4阶分圆数,构造了周期为p的完备高斯整数序列。利用序列积映射,可以构造更多参数的高斯完备序列。对于奇数长度的完备的高斯整数序列,利用奇偶变换,可以构造奇完备的高斯整数序列。这是首次构造了奇数长度的、完备和奇完备的高斯整数序列,首次给出了完备和奇完备QAM+序列的构造,肯定回答了Boztas和Parampalli关于完备QAM+序列存在性的问题。
本文利用交织技术和Gray映射的逆映射,研究了低/零奇相关区序列集的设计。基于交织技术,选取奇完备序列或者具有低奇周期自相关性质的序列作为列序列,利用已有的移位序列,提出了一种具有灵活参数的低/零奇相关区序列集的构造方法,得到了最优的低/零奇相关区序列集。另外,在适当选取移位序列和具有低奇自相关性质的二元序列,利用Gray映射的逆映射,构造了具有低奇相关区性质的四元序列集。基于已有的二元低奇相关区序列集,利用Gray映射的逆映射,设计了具有低奇相关性质的四元序列集。
接着本文对格雷序列和QAM格雷序列的进行了深入的研究,发现了部分四元格雷序列和QAM格雷序列具有完备和奇完备循环共轭性质。类似于格雷序列,设计了几类周期互补对和奇周期互补对。这些互补对可以用于设计最优的签名序列。
最后,本文研究了二元签名序列的理论界及其最优构造。注意到奇周期互相关函数和偶周期互相关函数的对称地位,本文给出了奇周期完全相关平方和的理论下界,并建立了达到理论界的签名序列与奇周期互补集的联系。从已知的奇完备的三元序列出发,构造了二元奇周期互补序列对。利用多项式函数,构造了一些新的二元奇周期互补集。除此之外,本文从m序列出发,构造了新的二元签名序列集,推广了Ganapathy、Pados和Karystinos基于Gold序列集的二元签名序列集的构造。
Sequences with low (odd) even periodic correlation have been applied in cryptography, code division multiple access communication systems, orthogonal frequency division multi-plexing communication systems, coding, Radar, Sonar and so on. In many communication systems, the performances are dominated by properties of sequences. In this thesis, we mainly focus on the study of sequences with low (odd) even periodic correlation in five topics:the bound of frequency hopping sequence sets and their optimal constructions, constructions of (almost) perfect and odd perfect sequences, constructions of low/zero odd correlation zone, the perfect and odd perfect cyclically conjugated property of Golay sequences and Golay QAM sequences, the perfect and odd perfect periodic complementary pairs and odd periodic complementary pairs which have similar form to Golay sequences, and the bound of binary signature sequences and their optimal constructions.
Firstly, according to the work given by Song et al. and Ding et al., we will further investigate the relationship between frequency hopping sequences and cyclic codes, and will discuss the relationship of several known bounds. We find the relationship of Peng-Fan bound and Plotkin bound on the frequency hopping sequences from the Plotkin bound in coding the-ory, the relationship of two Singleton bounds on frequency hopping sequences, which comes from the Singleton bound in coding theory, and the relationship of two Peng-Fan bounds. Furthermore, based on several classes of cyclic codes, such as Reed-Solomon codes and their punctured codes, codes defined by polynomial functions, and two classes of MDS codes and their punctured codes, we will design several classes of optimal frequency hopping sequence sets, which are optimal with respect to the new Singleton bound on the maximum Hamming correlation.
Secondly, we will study the construction of (almost) perfect and odd perfect sequences. Using the difference balanced functions, we will define new shift sequences, which gener-alizes the method to construct almost perfect and odd perfect ternary sequences given by Krengel. Choosing almost perfect sequences with balanced property, we can obtain (almost) perfect and odd perfect ternary sequences, Gaussian integer sequences, and QAM+sequences. We also design a class of almost perfect16-QAM sequences. For any odd prime p, we use the2and4order cyclotimic numbers with respect to p to construct perfect Gaussian integer sequences. Based on product mapping of sequences, more perfect and odd perfect Gaussian integer sequences can be obtained. By using even-odd transformation, odd perfect Gaussian integer sequences can also be derived. This thesis first constructs the perfect and odd perfect Gaussian integer sequences of odd length. and first presents perfect and odd perfect QAM+sequences, which positively answers the existence problem on perfect QAM+sequences pro-posed by Boztas and Parampalli.
In this thesis, based on interleaving technique and the inverse mapping of Gray mapping, we will study the construction of low/zero odd correlation zone sequence sets. According to interleaving technique, choosing odd perfect sequences or sequence with low odd autocorrela-tion property and using the known shift sequences, we will propose a construction of low/zero odd correlation zone sequences with flexible parameters. Those proposed sequences sets are optimal. Besides, applying the inverse mapping of Gray mapping to a binary sequence with low odd autocorrelation and choosing a proper shift sequence, we will construct quaternary sequence sets with low odd correlation zone property. Based on a known binary sequence set with low correlation zone property, using the inverse mapping of Gray mapping, we can obtain a quaternary sequence set with low odd correlation zone.
Next, we will study Golay sequences and QAM Golay sequences, and present that some quaternary Golay sequences and QAM Golay sequences have perfect and odd perfect cycli-cally conjugated property. Similar to Golay sequences, we give several classes of periodic complementary pairs and odd periodic complementary pairs, which can be used to construct optimal signature sequences.
Finally, we will study the bound of binary signature sequences and its optimal construc-tions. Note that odd periodic correlation function has the same importance as even periodic correlation function, we propose the lower bound of odd periodic total squared correlation, and obtain the relationship between and optimal signature sequences and odd periodic com-plementary sequence sets. We construct odd periodic complementary pair from known odd perfect ternary sequences. Using polynomial functions, we propose a class of new odd pe-riodic complementary sets. Besides, using m-sequences, we give new binary signature sets, which generalize the construction of binary signature sequences from Gold sequence sets given by Ganapathy, Pados and Karystinos.
首先,根据Song等人ODing等人的工作,本文进一步研究了跳频序列集和循环码之间存在密切的联系,讨论了已知的理论界之间的关系。发现了从Plotkin编码理论界得到的Peng-Fan界Plotkin跳频序列理论界的之间的联系、从Singleton编码理论界出发得到的两个跳频序列理论界之间的联系以及两个Peng-Fan界之间的关系。除此之外,基于一些的循环码,包括Reed-Solomon码以及其截短码、多项式函数定义的码、以及两类MDS码及其截短的MDS码,设计了达到新的最大汉明相关的Singleton界的最优跳频序列集。
其次研究了(几乎)完备和奇完备序列的构造。利用差平衡函数,定义了新的移位序列,推广了Krengel的几乎完备和奇完备三元序列的构造方法。选取平衡的几乎完备序列,构造了(几乎)完备和奇完备的三元序列、高斯整数序列以及QAM+序列,还设计了具有几乎完备性质的16-QAM序列。论文将利用素数p的2阶和4阶分圆数,构造了周期为p的完备高斯整数序列。利用序列积映射,可以构造更多参数的高斯完备序列。对于奇数长度的完备的高斯整数序列,利用奇偶变换,可以构造奇完备的高斯整数序列。这是首次构造了奇数长度的、完备和奇完备的高斯整数序列,首次给出了完备和奇完备QAM+序列的构造,肯定回答了Boztas和Parampalli关于完备QAM+序列存在性的问题。
本文利用交织技术和Gray映射的逆映射,研究了低/零奇相关区序列集的设计。基于交织技术,选取奇完备序列或者具有低奇周期自相关性质的序列作为列序列,利用已有的移位序列,提出了一种具有灵活参数的低/零奇相关区序列集的构造方法,得到了最优的低/零奇相关区序列集。另外,在适当选取移位序列和具有低奇自相关性质的二元序列,利用Gray映射的逆映射,构造了具有低奇相关区性质的四元序列集。基于已有的二元低奇相关区序列集,利用Gray映射的逆映射,设计了具有低奇相关性质的四元序列集。
接着本文对格雷序列和QAM格雷序列的进行了深入的研究,发现了部分四元格雷序列和QAM格雷序列具有完备和奇完备循环共轭性质。类似于格雷序列,设计了几类周期互补对和奇周期互补对。这些互补对可以用于设计最优的签名序列。
最后,本文研究了二元签名序列的理论界及其最优构造。注意到奇周期互相关函数和偶周期互相关函数的对称地位,本文给出了奇周期完全相关平方和的理论下界,并建立了达到理论界的签名序列与奇周期互补集的联系。从已知的奇完备的三元序列出发,构造了二元奇周期互补序列对。利用多项式函数,构造了一些新的二元奇周期互补集。除此之外,本文从m序列出发,构造了新的二元签名序列集,推广了Ganapathy、Pados和Karystinos基于Gold序列集的二元签名序列集的构造。
Sequences with low (odd) even periodic correlation have been applied in cryptography, code division multiple access communication systems, orthogonal frequency division multi-plexing communication systems, coding, Radar, Sonar and so on. In many communication systems, the performances are dominated by properties of sequences. In this thesis, we mainly focus on the study of sequences with low (odd) even periodic correlation in five topics:the bound of frequency hopping sequence sets and their optimal constructions, constructions of (almost) perfect and odd perfect sequences, constructions of low/zero odd correlation zone, the perfect and odd perfect cyclically conjugated property of Golay sequences and Golay QAM sequences, the perfect and odd perfect periodic complementary pairs and odd periodic complementary pairs which have similar form to Golay sequences, and the bound of binary signature sequences and their optimal constructions.
Firstly, according to the work given by Song et al. and Ding et al., we will further investigate the relationship between frequency hopping sequences and cyclic codes, and will discuss the relationship of several known bounds. We find the relationship of Peng-Fan bound and Plotkin bound on the frequency hopping sequences from the Plotkin bound in coding the-ory, the relationship of two Singleton bounds on frequency hopping sequences, which comes from the Singleton bound in coding theory, and the relationship of two Peng-Fan bounds. Furthermore, based on several classes of cyclic codes, such as Reed-Solomon codes and their punctured codes, codes defined by polynomial functions, and two classes of MDS codes and their punctured codes, we will design several classes of optimal frequency hopping sequence sets, which are optimal with respect to the new Singleton bound on the maximum Hamming correlation.
Secondly, we will study the construction of (almost) perfect and odd perfect sequences. Using the difference balanced functions, we will define new shift sequences, which gener-alizes the method to construct almost perfect and odd perfect ternary sequences given by Krengel. Choosing almost perfect sequences with balanced property, we can obtain (almost) perfect and odd perfect ternary sequences, Gaussian integer sequences, and QAM+sequences. We also design a class of almost perfect16-QAM sequences. For any odd prime p, we use the2and4order cyclotimic numbers with respect to p to construct perfect Gaussian integer sequences. Based on product mapping of sequences, more perfect and odd perfect Gaussian integer sequences can be obtained. By using even-odd transformation, odd perfect Gaussian integer sequences can also be derived. This thesis first constructs the perfect and odd perfect Gaussian integer sequences of odd length. and first presents perfect and odd perfect QAM+sequences, which positively answers the existence problem on perfect QAM+sequences pro-posed by Boztas and Parampalli.
In this thesis, based on interleaving technique and the inverse mapping of Gray mapping, we will study the construction of low/zero odd correlation zone sequence sets. According to interleaving technique, choosing odd perfect sequences or sequence with low odd autocorrela-tion property and using the known shift sequences, we will propose a construction of low/zero odd correlation zone sequences with flexible parameters. Those proposed sequences sets are optimal. Besides, applying the inverse mapping of Gray mapping to a binary sequence with low odd autocorrelation and choosing a proper shift sequence, we will construct quaternary sequence sets with low odd correlation zone property. Based on a known binary sequence set with low correlation zone property, using the inverse mapping of Gray mapping, we can obtain a quaternary sequence set with low odd correlation zone.
Next, we will study Golay sequences and QAM Golay sequences, and present that some quaternary Golay sequences and QAM Golay sequences have perfect and odd perfect cycli-cally conjugated property. Similar to Golay sequences, we give several classes of periodic complementary pairs and odd periodic complementary pairs, which can be used to construct optimal signature sequences.
Finally, we will study the bound of binary signature sequences and its optimal construc-tions. Note that odd periodic correlation function has the same importance as even periodic correlation function, we propose the lower bound of odd periodic total squared correlation, and obtain the relationship between and optimal signature sequences and odd periodic com-plementary sequence sets. We construct odd periodic complementary pair from known odd perfect ternary sequences. Using polynomial functions, we propose a class of new odd pe-riodic complementary sets. Besides, using m-sequences, we give new binary signature sets, which generalize the construction of binary signature sequences from Gold sequence sets given by Ganapathy, Pados and Karystinos.
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