Some bounds for the number of blocks III

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• 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 b=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16300656&_mathId=si1.gif&_user=111111111&_pii=S0012365X16300656&_rdoc=1&_issn=0012365X&md5=df7574c802f81b2e6a47be3840262cf3" title="Click to view the MathML source">D=(Ω,B)$\mathcal{D}=(\Omega ,\mathcal{B})$ be a pair of a b=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16300656&_mathId=si2.gif&_user=111111111&_pii=S0012365X16300656&_rdoc=1&_issn=0012365X&md5=8191f8c8b226115b5b7364001c326bcf" title="Click to view the MathML source">v$v$ point set b=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16300656&_mathId=si3.gif&_user=111111111&_pii=S0012365X16300656&_rdoc=1&_issn=0012365X&md5=d28645a9e3197231caf6cb828443b86e" title="Click to view the MathML source">Ω$\Omega$ and a set b=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16300656&_mathId=si4.gif&_user=111111111&_pii=S0012365X16300656&_rdoc=1&_issn=0012365X&md5=803140d191e5b01643bec65effd81984" title="Click to view the MathML source">B$\mathcal{B}$ consisting of b=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16300656&_mathId=si5.gif&_user=111111111&_pii=S0012365X16300656&_rdoc=1&_issn=0012365X&md5=7ceabda26ef7f9044530f07539959bfa" title="Click to view the MathML source">k$k$ point subsets of b=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16300656&_mathId=si3.gif&_user=111111111&_pii=S0012365X16300656&_rdoc=1&_issn=0012365X&md5=d28645a9e3197231caf6cb828443b86e" title="Click to view the MathML source">Ω$\Omega$ which are called blocks. Let b=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16300656&_mathId=si7.gif&_user=111111111&_pii=S0012365X16300656&_rdoc=1&_issn=0012365X&md5=0fac827bbaff29e89a4d1b9e25f3bb48" title="Click to view the MathML source">d$d$ be the maximal cardinality of the intersections between the distinct two blocks in b=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16300656&_mathId=si4.gif&_user=111111111&_pii=S0012365X16300656&_rdoc=1&_issn=0012365X&md5=803140d191e5b01643bec65effd81984" title="Click to view the MathML source">B$\mathcal{B}$. The triple b=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16300656&_mathId=si9.gif&_user=111111111&_pii=S0012365X16300656&_rdoc=1&_issn=0012365X&md5=c633793992b75ca51d5730a0b8b1c5e2" title="Click to view the MathML source">(v,k,d)$(v,k,d)$ is called the parameter of b=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16300656&_mathId=si4.gif&_user=111111111&_pii=S0012365X16300656&_rdoc=1&_issn=0012365X&md5=803140d191e5b01643bec65effd81984" title="Click to view the MathML source">B$\mathcal{B}$. Let b=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16300656&_mathId=si11.gif&_user=111111111&_pii=S0012365X16300656&_rdoc=1&_issn=0012365X&md5=f89db42a1e9be0243ec171898c461172" title="Click to view the MathML source">b$\mathrm{<font\; color="red">b</font>}$ be the number of the blocks in b=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16300656&_mathId=si4.gif&_user=111111111&_pii=S0012365X16300656&_rdoc=1&_issn=0012365X&md5=803140d191e5b01643bec65effd81984" title="Click to view the MathML source">B$\mathcal{B}$. It is shown that inequality b=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16300656&_mathId=si13.gif&_user=111111111&_pii=S0012365X16300656&_rdoc=1&_issn=0012365X&md5=fed898389ed350ff14c4b3db7eaf1b1c">$\left(\genfrac{}{}{0ex}{}{v}{d+2i-1}\right)\ge \mathrm{<font\; color="red">b</font>}\{\left(\genfrac{}{}{0ex}{}{k}{d+2i-1}\right)+\left(\genfrac{}{}{0ex}{}{k}{d+2i-2}\right)\left(\genfrac{}{}{0ex}{}{v-k}{1}\right)+\cdots +\left(\genfrac{}{}{0ex}{}{k}{d+i}\right)\left(\genfrac{}{}{0ex}{}{v-k}{i-1}\right)\}$ holds for each b=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16300656&_mathId=si14.gif&_user=111111111&_pii=S0012365X16300656&_rdoc=1&_issn=0012365X&md5=0b89e1b868929ac0f2bc301c74777f73" title="Click to view the MathML source">i$i$ satisfying b=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16300656&_mathId=si15.gif&_user=111111111&_pii=S0012365X16300656&_rdoc=1&_issn=0012365X&md5=6028ebd7f791b1c5dc624ce671d75f4f" title="Click to view the MathML source">1≤i≤k−d$1\le i\le k-d$, in the paper Noda (2001).

If b=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16300656&_mathId=si11.gif&_user=111111111&_pii=S0012365X16300656&_rdoc=1&_issn=0012365X&md5=f89db42a1e9be0243ec171898c461172" title="Click to view the MathML source">b$\mathrm{<font\; color="red">b</font>}$ achieves the upper bound for some b=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16300656&_mathId=si14.gif&_user=111111111&_pii=S0012365X16300656&_rdoc=1&_issn=0012365X&md5=0b89e1b868929ac0f2bc301c74777f73" title="Click to view the MathML source">i$i$, b=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16300656&_mathId=si15.gif&_user=111111111&_pii=S0012365X16300656&_rdoc=1&_issn=0012365X&md5=6028ebd7f791b1c5dc624ce671d75f4f" title="Click to view the MathML source">1≤i≤k−d$1\le i\le k-d$, then b=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16300656&_mathId=si19.gif&_user=111111111&_pii=S0012365X16300656&_rdoc=1&_issn=0012365X&md5=7191d9c816b39605ed1daa3e1dd0c3b9" title="Click to view the MathML source">D$\mathcal{D}$ is called a b=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16300656&_mathId=si20.gif&_user=111111111&_pii=S0012365X16300656&_rdoc=1&_issn=0012365X&md5=d03dc44018b6e0810042ca9483baf51e" title="Click to view the MathML source">β(i)$\beta \left(i\right)$ design. In the paper mentioned above, an upper bound and a lower bound, b=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16300656&_mathId=si21.gif&_user=111111111&_pii=S0012365X16300656&_rdoc=1&_issn=0012365X&md5=420e351da8dd33cffb95679244a38a5b">$\frac{(d+2i)(k-d)}{i}\le v\le \frac{(d+2(i-1))(k-d)}{i-1}$, for b=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16300656&_mathId=si2.gif&_user=111111111&_pii=S0012365X16300656&_rdoc=1&_issn=0012365X&md5=8191f8c8b226115b5b7364001c326bcf" title="Click to view the MathML source">v$v$ of b=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16300656&_mathId=si20.gif&_user=111111111&_pii=S0012365X16300656&_rdoc=1&_issn=0012365X&md5=d03dc44018b6e0810042ca9483baf51e" title="Click to view the MathML source">β(i)$\beta \left(i\right)$ design b=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16300656&_mathId=si19.gif&_user=111111111&_pii=S0012365X16300656&_rdoc=1&_issn=0012365X&md5=7191d9c816b39605ed1daa3e1dd0c3b9" title="Click to view the MathML source">D$\mathcal{D}$ are given. In this paper we consider the cases when b=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16300656&_mathId=si2.gif&_user=111111111&_pii=S0012365X16300656&_rdoc=1&_issn=0012365X&md5=8191f8c8b226115b5b7364001c326bcf" title="Click to view the MathML source">v$v$ does not achieve the upper bound or lower bound given above, and get new more strict bounds for b=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16300656&_mathId=si2.gif&_user=111111111&_pii=S0012365X16300656&_rdoc=1&_issn=0012365X&md5=8191f8c8b226115b5b7364001c326bcf" title="Click to view the MathML source">v$v$ respectively. We apply this bound to the problem of the perfect b=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16300656&_mathId=si27.gif&_user=111111111&_pii=S0012365X16300656&_rdoc=1&_issn=0012365X&md5=5ccc027ec29475af585d91d45b841b45" title="Click to view the MathML source">e$e$-codes in the Johnson scheme, and improve the bound given by Roos in the paper Roos (1983).