On The Existence Of Perfect Difference Families And Related Problems

Perfect difference family is an important kind of combinatorial designs,and it can be used to construct cyclic Steiner 2-design C(v,k,1).It has close relationship with perfect system of difference sets,which have been applied in the significant fields of radioastronomy, graph theory,experimental design,digital communication and computer coding,etc.In this thesis,the existence of perfect difference families,directed perfect difference families and perfect systems of difference sets with certain parameters and other related problems are investigated.Some new existence results on(gv,g,k,1)-directed perfect difference families in Z_gv,(gv,g,3,1)-directed perfect difference families in Z_gv and(m,k,c)-systems(k=4,5)are presented.There are five chapters in this thesis.In Chapter 1,we introduce some basic definitions and some known conclusions of perfect difference family,directed perfect difference family,perfect system of difference sets,additive sequence of permutations,(m,k,c)-system and its split,etc.In Chapter 2,a necessary condition for the existence of(gv,g,k,1)-directed perfect difference families in Z_gvis obtained.In Chapter 3,the necessary conditions for the existence of(gv,g,3,1)-directed perfect difference families in Z_gvare given,and some existence results on(gv,g,3,1)-directed perfect difference families in Z_gvare shown.In Chapter 4,the necessary conditions for the existence of(m,k,c)-systems are shown and some known existence results are listed.Besides,applying some known results and the related contents mentioned in Chapter 1,several new results on(m,k,c)-systems(k=4,5)are obtained by direct constructions and recursive constructions.In Chapter 5,the main results of this thesis are summarized,and finally the further research problems are presented.