Some Bounds for the Number of Blocks III

Abstract

Let $\mathcal{D}=(\Omega ,\mathcal{B})$ be a pair of $v$ point set $\Omega$ and a set $\mathcal{B}$ consists of $k$ point subsets of $\Omega$ which are called blocks. Let $d$ be the maximal cardinality of the intersections between the distinct two blocks in $\mathcal{B}$. The triple $(v,k,d)$ is called the parameter of $\mathcal{B}$. Let $b$ be the number of the blocks in $\mathcal{B}$. It is shown that inequality $\left(\genfrac{}{}{0ex}{}{v}{d+2i-1}\right)\ge b\{(\genfrac{}{}{0ex}{}{k}{d+2i-1})+(\genfrac{}{}{0ex}{}{k}{d+2i-2}\left)\right(\genfrac{}{}{0ex}{}{v-k}{1})+....$ $.+\left(\genfrac{}{}{0ex}{}{k}{d+i}\right)\left(\genfrac{}{}{0ex}{}{v-k}{i-1}\right)\}$ holds for each $i$ satisfying $1\le i\le k-d$, in the paper: Some Bounds for the Number of Blocks, Europ. J. Combinatorics 22 (2001), 91--94, by R. Noda. If $b$ achieves the upper bound, $\mathcal{D}$ is called a $\beta (i)$ design. In the paper, an upper bound and a lower bound, $\frac{(d+2i)(k-d)}{i}\le v\le \frac{(d+2(i-1))(k-d)}{i-1}$, for $v$ of a $\beta (i)$ design $\mathcal{D}$ are given. In the present paper we consider the cases when $v$ does not achieve the upper bound or lower bound given above, and get new more strict bounds for $v$ respectively. We apply this bound to the problem of the perfect $e$-codes in the Johnson scheme, and improve the bound given by Roos in 1983.

Stylistic corrections are made. References are added