The Constructions Of Difference Set Pairs And Sequence Pairs Based On Cyclotomy

The perfect discrete signal is an array signal with good autocorrelation properties. Itis applied in many areas of engineering and science such as modern communication, radar,sonar and so on. However, according to the results of the current study, the existenceconditions and the application mode of the perfect sequences restrict the amount of it, andfurther make it can not meet the actual needs of the project. It leads exporing andproposing a new sense of perfect signal form becoming more and more urgent. Thus, theconcept of " pairs " have come into being. The design and constructions of sequence pairsbecomes an important breakthrough in the direction of constructing more and more idealsequence. And the difference set pairs is, precisely in order to study the new form idealsequence—sequence pairs, presented as a mathematical tool. Cyclotomic class as one ofthe three main tools to study the difference set pairs, their research in terms of thedifference set pairs’ construction is not perfect, yet deeper discussion. So in this paper, wemainly study the constructions of the difference set pairs and the sequence pairs based oncyclotomic class.Firstly, proposed the difference set pairs search algorithms and the sequence pairssearch algorithms based on cyclotomy, get amount of experimental data, which lay afoundation for proposing new constructions of difference set pairs and sequence pairs. Inorder to facilitate the further study of difference set pairs and sequence pairs, as well as theinquiry management of its related data instances, the difference set pairs and sequencepairs data management system based on cyclotomy is designed and implemented. Thesystem assemble the function of the data instances search, query and maintenance, and thetheorem query, validating and addition in one. Many known constructing methods fordifference set pairs based on cyclotomy and two constructing methods for ideal sequencepairs are logged in the system, and meanwhile a interface is left in the system, in order tofacilitate the additions of the new constructors.Secondly, by studying the concept and the nature of difference set pairs andcyclotomic classes, as well as analysising a large mount of difference set pairs instances obtained by experiment, four new methods for constructing difference set pairs based oncyclotomy are proposed, and some new difference set pairs instance on Abel group oforder13are given out. Those constructing methods enrich the theoretical research ondifference set pairs, and lay a theoretical foundation for its further study. At the same time,they provide a strong mathematical support for the constructing of more perfect binaryarray pairs.Finally, by analysising a large mount of statistical data of sequence pairs’ relationcharacteristics, which obtained by experiment, two new methods for constructing idealternary sequence pairs based on cyclotomy are presented. One of them, which based oncyclotomic numbers of order4, gives out ternary sequence pairs with three-levelcorrelation. And the other ternary sequence pairs with four-level correlation areconstructed based on cyclotomic numbers of order6. Both of the methods cover all theprime period of order4and6respectively. The two constructing methods not only enrichthe theoretical research on sequence pairs, but also provide more available ideal signals forthe actual project needs.