The Study Of Sequences With Low(ODD) Even Correlation

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.