Preliminary Study On Skew Difference Families

Difference families and almost difference sets are two kinds of important combinatorial designs. They have been applied in cryptography, coding. theory and binary sequence. For example, they can be used to construct cryptographic functions with optimal nonlinearity, binary sequences with optimal autocorrelation and good constant weight codes, etc.In this thesis, the existence of skew difference families and almost difference sets with special parameters are investigated, and some relationships between difference families and skew difference families are established.There are three chapters in this thesis.In Chapter 1, we introduce the basic definition of difference set, almost difference set. difference family and cyclotomic class. Some properties about cyclotomic numbers are obtained, which are useful in Chapter 2.In Chapter 2, a survey is given on the known results of skew difference families. Then the existence of a (q, (q-1)/2, (q-3)/2)-skew difierence family is reproofed for q≡5 (mod 8) by the method of cyclotomic class, and more results are presented on this kind of skew difference families with q≡1, 5 (rood 8).Furthermore, a new class of skew almost difference sets with parameter (q, (q- 1)/2, (q-5)/4, (q-1)/2) is obtained, and some results on this kind of skew almost difference sets are listed with computer search.At the end of this chapter, we show the connection between difference families and skew difference families. Some results on non-existence of skew difference families are also listed.In Chapter 3, the main conclusions of this thesis are summarized, and the further research problems are presented at last.