Some bounds for the number of blocks III

详细信息    查看全文

• 作者：Etsuko Bannai^a ; ^{et-ban@rc4.so-net.ne.jp" class="auth_mail" title="E-mail the corresponding author} ; Ryuzaburo Noda^b ; ^{rrnoda@yahoo.co.jp" class="auth_mail" title="E-mail the corresponding author}
• 关键词：β(i)β(i) design ; β(i)β(i) set ; Johnson scheme ; Perfect ee-code ; Diameter perfect code
• 刊名：Discrete Mathematics
• 出版年：2016
• 出版时间：6 September 2016
• 年：2016
• 卷：339
• 期：9
• 页码：2313-2328
• 全文大小：447 K

文摘

Let D=(Ω,B)$\mathcal{D}=(\Omega ,\mathcal{B})$ be a pair of a v$v$ point set Ω$\Omega$ and a set B$\mathcal{B}$ consisting of k$k$ point subsets of Ω$\Omega$ which are called blocks. Let d$d$ be the maximal cardinality of the intersections between the distinct two blocks in B$\mathcal{B}$. The triple (v,k,d)$(v,k,d)$ is called the parameter of B$\mathcal{B}$. Let b$b$ be the number of the blocks in B$\mathcal{B}$. It is shown that inequality $\left(\genfrac{}{}{0ex}{}{v}{d+2i-1}\right)\ge b\{\left(\genfrac{}{}{0ex}{}{k}{d+2i-1}\right)+\left(\genfrac{}{}{0ex}{}{k}{d+2i-2}\right)\left(\genfrac{}{}{0ex}{}{v-k}{1}\right)+\cdots +\left(\genfrac{}{}{0ex}{}{k}{d+i}\right)\left(\genfrac{}{}{0ex}{}{v-k}{i-1}\right)\}$ holds for each i$i$ satisfying 1≤i≤k−d$1\le i\le k-d$, in the paper Noda (2001).

If b$b$ achieves the upper bound for some i$i$, 1≤i≤k−d$1\le i\le k-d$, then D$\mathcal{D}$ is called a β(i)$\beta \left(i\right)$ design. In the paper mentioned above, 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$v$ of β(i)$\beta \left(i\right)$ design D$\mathcal{D}$ are given. In this paper we consider the cases when v$v$ does not achieve the upper bound or lower bound given above, and get new more strict bounds for v$v$ respectively. We apply this bound to the problem of the perfect e$e$-codes in the Johnson scheme, and improve the bound given by Roos in the paper Roos (1983).