The Existence Of Balanced Difference Families And Perfect Difference Families

In digital communication,such as global positioning system,code division multiple access(CDMA)systems for mobile communications,radar system and encryption of data, binary sequences with optimal two or three level autocorrelation play important role.These sequences are closely related to the difference families,balanced difference families and the perfect difference families.In 1939,R.C.Bose introduced the(v,k,λ)-difference families((v,k,λ)-DF in short) to construct(v,k,λ)-balanced incomplete block design(BIBD).In 1999,M.Buratti introduced balanced(v,K,λ)-difference families.Given an additive group G of order v,a set K of positive integers and a positive integer h.Let B be a family of B_i = {h_i1,b_i2,…,b_(ik)i},where B_i(?)G,k_i∈K and 1≤i≤h. For a given subset B of G,denoteâ–³B= {a - b∶a,b∈B,a≠b},â–³B=∪_{1≤i≤h}â–³B_i.Ifâ–³B=λ(G \ {0}),then B is called a difference family over G,denoted by(v,K,λ)-DF,B_i is called a base block and k_i is the length of the B_i.B is balanced if 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≤h.For the existence of balanced(v,K,λ)-DFs,the following result is given by M.Buratti when K= {3,4}:Lamma 1.2.2 Let v= 18t+1 be a prime power and let 3^e be the highest power of 3 dividing t.Then,if 3 is not a 3^e+1th power in F_v,there exists a balanced(v,{3,4},1)-DF.It is not difficult to see that the necessary conditions for the existence of balanced (v,{3,4},λ)-DFs are as follows.(1)v≡1(mod 18),v≥19 ifλ≡1,5,7,11,13,17(mod 18);(2)v≡1(mod 9),v≥10 ifλ≡2,4,8,10,14,16(mod18);(3)v≡1(mod 6),v≥7 ifλ≡3,15(mod 18);(4)v≡1(mod 3),v≥4 ifλ≡6,12(mod 18);(5)v≡1(mod 2),v≥4 ifλ≡9(mod 18);(6)v≥4 ifλ≡0(mod 18). By using Weil's theorem on multiplication character sum estimates,an explicit lower bound for tile existence of(v,{3,4},λ)-DFs is obtained forλ=1,3.With the aid of a computer,the following result is obtained:Theorem 1.4.1 If v is a prime power,then the necessary conditions for the existence of balanced(v,{3,4},λ)-DFs are also sufficient.Another work of this paper is focus on perfect difference families.R.J.R.Abel,R. Mathon and A.Rosa et al had used perfect difference families to solve the problems of the cyclic steiner 2-design,optical orthogonal codes and graph theory.Recently,G.Ge,A.C.H.Ling and Y.Miao generalized the concept of perfect difference family and used it to construct radar array and PCPM(Properly Centered Permutation Matrices).A partial answer to the open problem on PCPM posed by Z.Zhang and C.Tu was also given.Let F={B₁,B₂,...,B_h},where B_i= {b_il,b_i2,...,b_(ik)i},K={k₁,k₂,...,k_h} be a collection of h subsets of I_v={0,1,...,v - 1} called blocks.If the differencesâ–³+F= {b_in-b_im∶1≤i≤h,1≤m＜n≤k_i} cover the set {1,2,...,(v-1)/2},then F is called a perfect(v,K,1)-difference family over I_v,or briefly,a perfect(v,K,1)-DF,k_i is called the length of the B_i.When K={k},the notation perfect(v,k,1)-DF is used.A perfect (v,{w,s₁^*,s₂^*,s₃^*,...s_t^*},1)-DF is defined to be a perfect(v,{w,s₁,s₂,...s_t},1)-DF with exactly one block of size s_j,1≤j≤t,the size of the rest blocks is w,where w,S₁,s₂,...,s_t are pair different positive integers.For the existence of perfect(v,K,1)-DF,partial results were obtained by R.Mathon et al when K= {k}.While little is know for perfect(v,K,1)-DF when |K|＞1.In this paper,a recursive construction of one kind of perfect difference family is obtained by using of Langford sequence,and a nonexistence result is also given.Next,the existence of K = {3,s^*} and K = {3,s₁^*,s₂^*} are obtained.Finally,further discussions and new results of recursive constructions of(v,k,1)-DF are given.The main results of this thesis are as follows:Theorem 1.4.2 Suppose that there exists a perfect(v,K,1)-DF,then there exists a perfect(6m+v,K∪{3},1)-DF if the following two conditions hold:(1)m≥v;(2)m≡0,1(mod 4)when v≡1(mod 4);m≡0,3(mod 4)when v≡3(mod 4).Theorem 1.4.3 There exists no perfect(v,{3,s^*},1)-DF,if one of the following conditions holds:(1)v- r≡13,19(mod 24)when s≡1(mod 8);(2)v- r≡7,13(mod 24)when s≡3(mod 8);(3)v- r≡1,7(mod 24)when s≡5(mod 8);(4)v- r≡1,19(mod 24)when s≡7(mod 8); where r=s(s-1).Theorem 1.4.4 There exists no perfect(v,K,1)-DF,if min {k∶k∈K}≥6.Theorem 1.4.5 For the following K and v,each condition is necessary and sufficient for the existence of a perfect(v,K,1)-DF(1)K={3,4^*},v≡1(mod 6),v≥19;(2)K={3,5^*},v≡9,15(mod 24),v≥33;(3)K={3,6^*},v≡1(mod 6),v≥43;(4)K={3,7^*},v≡1,7(mod 24),v≥73Theorem 1.4.6 For K={3,4^*,5^*},{3,4^*,6^*},{3,5^*,6^*},the necessary condition of the existence for a perfect(v,K,1)-DF is also sufficient.Theorem 1.4.7 Suppose a perfect(12t+1,4,1)-DF exists,then perfect(L,4,1)-DF exists for L=324t+61,324t+73,348t+73,348t+85.Theorem 1.4.8 If there exist both a perfect(v,k,1)-DF and a perfect(w,k,1)-DF then a there exists a perfect(vw,k,1)-DF for k=4,5.