Difference Set Pairs And Almost Difference Set Pairs Based On Generalized Cyclotomic Classes

Ideal sequences and sequence pairs are widely used in many fields such as information communication systems,electronic technology,and cryptography.However,it is difficult to construct the ideal sequence and the ideal sequence pairs directly.Therefore,many scholars will use some tools to construct ideal sequence pairs,such as difference set pairs and almost difference set pairs.Cyclotomic class is a classical theory in finite field,but its application is limited.Scholars use the Chinese remainder theorem and the generalized Chinese remainder theorem to generalize the cyclotomic class on the ring to the generalized cyclotomic class.Cyclotomic class and generalized cyclotomic class are usually applied to the construction of difference set,almost difference set,difference set pairs and almost difference set pairs.Difference set pairs and almost difference set pairs are the most direct tools for constructing ideal sequence pairs,and corresponding construction results have also been proposed.In the past,scholars mainly used classical cyclotomic class and Whiteman generalized cyclotomic class to construct difference set pairs and almost difference set pairs.In this paper,we propose a conjecture based on Z_2p^m generalized cyclotomic class and Fan-Ge generalized cyclotomic class to construct several sets of difference set pairs and almost difference set pairs with new parameters.The main contents are as follows:Firstly,the article introduces the definition of the classic cyclotomic class,and several different definitions after it is generalized to the generalized cyclotomic class on the ring.At the same time,the research situation of difference set pairs and almost difference set pairs and some research results are introduced.And the definition and examples of difference set pairs and almost difference set pairs are given,which lays the theoretical foundation for the subsequent conjecture of constructing difference set pairs and almost difference set pairs based on the generalized cyclotomic class.Secondly,through the study and research of the Z_2p^m generalized cyclotomic class and the Z_2p^m generalized cyclotomic number.Since the Z_2p^m generalized cyclotomic number is somewhat different from the classical generalized cyclotomic number,the result of the Z_2p^m generalized cyclotomic number obtained by the difference of the two set elements is more complicated.In this article,twenty-four typical situations are described,and the number of occurrences of various elements is listed in detail.Through the research on the method of dividing circle construction of difference set pairs and almost difference set pairs,the conjectures for constructing several sets of new difference set pairs and almost difference set pairs are proposed,and a large amount of data is verified through experiments.Finally,by using the definition of Fan-Ge generalized cyclotomic class,we propose four conjectures of almost difference set pairs.Based on the study of the definition of Zv(v=p1e1p2e2…pnen)generalized cyclotomic class,several groups of different group pair sets are found by using the properties of irreversible elements in rings.At the same time,through a large number of data analysis,the conjecture of constructing several groups of almost difference set pairs with parameters(v,k,k',e,?,t) on Z_v are proposed.