Several Constructions Of Variants Of Difference Set On Galois Ring GR(p²,p^2s)

As a promotion of finite fields,Galois ring has good properties and has attracted more and more scholars’ attention.It has been widely used in the fields of coding theory,communication and information security.Difference sets and difference families are an important research direction in combinatorial mathematics.Trying to construct difference sets and difference families on Galois rings has always been a research hotspot.This paper explores the construction of several difference sets deformation problems on Galois rings.Firstly,it introduces the background knowledge of Galois ring and various difference sets,difference families,etc.,including the role of Galois ring theoretical research in encrypted communication systems,the importance of algebraic research,and the progress of background knowledge related to this paper.Secondly,the basic concepts and properties required for the thesis are explained.It is divided into two parts:One is the basic knowledge of algebra,including the basic theories of groups,rings,and fields,finite fields and trace mapping,and the definition of difference family,almost difference family,and divisible difference set;The second is the basic elements of Galois rings.Finally,starting from the Galois ring GR(p2,p2s),we explore three variants of difference set.When the parameter p=2,for a disjoint difference family on the Galois ring GR(22,22s),by adding an element to a certain set in the difference family,when a certain condition is met,one almost difference family on GR(22,22s)can be constructed.In the experiment,it is found that when the added element is 1,the result meets the almost difference family condition.When the parameters p and s take any positive integer,by finding the existing bijective between the ring and its subring,we construct sets Di.Then the number of reversible elements and that of irreversible elements in the difference of Di have a certain pattern.So,we give a conjecture,which is verified by a large number of experiments.We also consider the problem of the image of the trace mapping acting on the preimage of the trace mapping,and propose a sufficient condition that the sum of the images of a multiplicative character on a set is zero.Besides,we also try to change the function value corresponding to the trace mapping on the finite field.When the parameter p=3,we propose a conjecture of divisible difference set.In this paper,three kinds of difference set deformations on the Galois ring are constructed,and the construction results are verified by programming experiments.Some conjectures have not been proved,and the research needs to be continued in the next step.