The Existence Of Balanced Difference Families

In 1939, Bose introduced (v, k,λ) difference families((v, k,λ)-DF in short) to construct (v,k,λ) balanced incomplete block designs((v, k,λ)-BIBDs in short). A lot of results have been obtained for the existence of (v, k,λ)-DFs, while not so much is known when k is changed into a positive integer set K and |K| > 1, one reason is that the set of blocks is more complicate. Marco Buratti introduced balanced difference families (Balanced DF in shot). A few fesults on balanced DF via finite fields were also presented in.Given an additive group G of order v and a set K of positive integers. Given a family S which consists of B_i = {b_i1,b_i2,…,b_iki}, where B_i (?) G, 1≤i≤s. If the following conditions are satisfied:(1) non-zero element of G can be represented as the difference of two elements of some base block in exactly A ways:a = b_ij - b_il, 1≤i≤s, 1≤j, l≤k_i, j≠l;(2) k_i∈K, 1≤i≤s.Then S is called a (v, K,λ)-DF over G. We say that S is balanced, provided that K contains at least two distinct elements and the number of base blocks of size k_i is constant for each k_i∈K, 1≤i≤s.In this paper, the following construction is obtained:Theorem 1.1 Let e, r, s, k_i andλbe all positive integers, where 1≤i≤r + sand e is an odd. When m = 1/2λ[sum form i=1 to r k_i(ek_i-1)+sum form j=1 to s k_r+j(ek_r+j+1)] is also an integer. If v = 2emt + 1 is a prime power, then, letωbe a primitive root on F_v,εbe an eth primitive root of unity on F_v, A_i={a₁ⁱ,a₂ⁱ,…,a_kiⁱ}<ε>,A_r+j={a₁^r+j,a₂^r+j,…,a₍k_r+j^r+j}<ε>∪{0}be subset of F_v, where 1≤i≤r, 1≤j≤s. Then, if the listL=(a_d^r+j|1≤j≤s,1≤d≤k_r+j)∪(a_dⁱ(ε^h-1)|1≤i≤r+s,1≤d≤k_i,1≤h≤(e-1)／2)∪(a_d1ⁱ-a_d2ⁱε^h|1≤i≤r+s,1≤d₁2≤k_i,0≤h≤e-1), is evenly distributed over the mth power cosets of F_v^*, we have that the familyF= (A_iω^mj|1≤j≤t, 1≤i≤r + s) is a balanced (v, {ek₁,…, ek_r,ek_r+1 + 1,…, ek_r+s + 1},λ)-DF.Applying Theorem 1.1 with Weil's theorem on Character sum estimation, the following results are also obtained.Theorem 1.2 If v is a prime power, then the necessary conditions for the existence of a balanced (v, {3, 6},λ)-DF over F_v are also sufficient.Theorem 1.3 If v is a prime power, then the necessary conditions for the existence of a balanced (v, {3, 7}, 2)-DF over F_v are also sufficient.The paper is divided into four parts. In chapter one, we present some notions, the known results on (balanced) difference families and the main results of this paper. Chapter two talks about the existence of the balanced (v, {3,6},λ)-DFs. Chapter three discusses the existence of the balanced (v, {3, 7}, 2)-DFs. Further research problems are presented in Chapter four.