Recursive Constructions Of Difference Families On Direct Product Of Groups

Difference family is one of the most important concepts in design theory. It gives a brief account on the constructions and existence of BIB designs. Many methods for construction rely on the algebraic structure. Today, it has very fruitful connections with group theory,number theory and finite geometry. Let (u,k,1) difference family denote a difference family of order v with block size k in finite group. In this paper, we give a simple recursive construction of difference family on finite group.when k is a prime power.This paper is organized as follows.In the first chapter, we give a brief introduction on the background and main result of BIB design.In the second chapter, we recall some basic notions of difference family and orthogonal Latin square. And present two classes of different constructions of difference familyIn the third chapter, we concentrating on the constructions of difference family First, we talk about the problems of recursive constructions for difference family with k=3and k=4. Second, We obtained a (uv,k,1) difference family by applying the theory of orthogonal Latin square from (u,k,1) difference family and (v.k,1) difference family, when k is a prime power. Lastly, when k is not prime power, under the premise of the prime power conjecture, it is not difficult to check that the construction presented before can not be applied here.In the forth chapter, the equivalency of BIB design is considered. Firstly,an equiva-lence relation is established between the incidence matrix of BIB design and linear code Secondly, the minimum distribution of three [392,49]-linear codes are given with the aid of computer. At the end. we get the result that none of these three design of B(3,1,49) is equal to another.In the last chapter, we summarized the main results of this paper and presented some further problems.