The Constructions And Applications Of Zero-difference Balanced Functions

Zero-difference balanced function(ZDBF)is an optimal combinatorial object pro-posed by Ding in constructing the optimal constant composition codes(CCCs)in 2008.It was a generalization of the perfect nonlinear functions and be used in the constructions of difference system of sets.Zero-difference balanced functions are related to many subjects such as combinatorics,algebra,and finite geometry,etc,and have many applications in many fields such as coding theory,cryptography,and communication engineering,etc.In combination design theory,Ding pointed out a correspondence between zero-difference balanced function and partitioned difference family,which implies that each ZDBF can be identified with a PDF with parameter(n,{?0,?1,…,?l-1?,?).Since the parameter ??0,?1;…,?l-1} is not available in some cases,we also associate a ZDBF with a tuple of three parameters(n,l,?).This paper summarizes two methods of constructing the ZDBFs.One is the direct construction method using the cyclotomy theory and related known results,and the other is the recursive construction using the difference matrix and the known PDFs.Many new ZDBFs are given by using these two methods.Based on this,new constant composition codes,optimal and perfect difference system of sets,optimal frequency-hopping sequences are obtained using the ZDBFs with new parameters.The overall layout of this article is as follows.In the first chapter,the basic cyclotomy theory of finite fields,some combination concepts and known conclusions are given.In the second chapter,many new ZDBFs are obtained from the corresponding relationship between ZDBF and PDF,and many new results about a special ZDBF with zero as a single point set are obtained.In the third chapter,we summarize the applications of ZDBFs in optimal constant composition codes,perfect difference system of sets and optimal frequency-hopping sequences.