Existence Of Optimal Strong Partially Balanced 3-Designs With Block Size Four

Let v,k,λ,t be positive integers with t≤k. A partially balanced t-design is a pair(X,B)where X is a set of size v and B is a set of k-subsets (called blocks)of X such that any t-subset of X either occurs in exactlyλblocks or does not occur in any block. If(X,B)is also a partially balanced r-design for any integer 1≤r≤t-1 as well,then it is called a strong partially balanced t-design. Further, if there does not exist a strong partially balanced t-design(X,A)with |A|>|B|,then it is called an optimal strong partially balanced t-design.Strong partially balanced t-designs can be used to construct authentication codes, whose probabilities of successful deception in an optimum spoofing attack of order r for 0≤r≤t-1,achieve their information-theoretic lower bounds.Forλ=1,the spectra of strong partially balanced 2-designs with k=3,4,5 have been determined by Du Beiliang.In this thesis,we obtain an upper bound for the number of blocks of the strong partially balanced 3-design with k=4.With the help of candelabra quadruple system,we obtain a recursive construction for strong partially balanced 3-designs.Then we generalize the concept of matching CQS,together with s-fan designs we determine the existence of CQS(gn:s)for 9∈{6,12},s∈{2i:i∈Z/2+1}.Finally, we prove the following conclusion:let v,λbe integers,if(v,λ)(?){(a,b):a=12k+11,k∈(?),b三1(mod 2)}∪{(a,b):a=6k+5,k∈(?),b三2(mod 4)}, then there exists an optimal strong partially balanced 3-(v,4,λ,O)design,where (?)={m:m is odd, m∈[3,35]∪[41,55]∪[75,79]∪[159,175],m≠43}.