Some bounds for the number of blocks III

Let D=(Ω,B)D=(Ω,B) be a pair of a vv point set ΩΩ and a set BB consisting of kk point subsets of ΩΩ which are called blocks. Let dd be the maximal cardinality of the intersections between the distinct two blocks in BB. The triple (v,k,d)(v,k,d) is called the parameter of BB. Let bb be the number of the blocks in BB. It is shown that inequality vd+2i−1≥b{kd+2i−1+kd+2i−2v−k1+⋯+kd+iv−ki−1} holds for each ii satisfying 1≤i≤k−d1≤i≤k−d, in the paper Noda (2001).If bb achieves the upper bound for some ii, 1≤i≤k−d1≤i≤k−d, then DD is called a β(i)β(i) design. In the paper mentioned above, an upper bound and a lower bound, (d+2i)(k−d)i≤v≤(d+2(i−1))(k−d)i−1, for vv of β(i)β(i) design DD are given. In this paper we consider the cases when vv does not achieve the upper bound or lower bound given above, and get new more strict bounds for vv respectively. We apply this bound to the problem of the perfect ee-codes in the Johnson scheme, and improve the bound given by Roos in the paper Roos (1983).