{
  "id": "1404.3821",
  "version": "v2",
  "published": "2014-04-15T05:55:37.000Z",
  "updated": "2015-03-18T08:40:38.000Z",
  "title": "Some Bounds for the Number of Blocks III",
  "authors": [
    "Etsuko Bannai",
    "Ryuzaburo Noda"
  ],
  "comment": "Stylistic corrections are made. References are added",
  "categories": [
    "math.CO"
  ],
  "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 ${v\\choose d+2i-1}\\geq b\\{{k\\choose d+2i-1} +{k\\choose d+2i-2}{v-k\\choose 1}+....$ $.+{k\\choose d+i}{v-k\\choose i-1} \\}$ holds for each $i$ satisfying $1\\leq i\\leq k-d$, 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}\\leq v \\leq \\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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-04-15T05:55:37.000Z",
      "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 ${v\\choose d+2i-1}\\geq b\\left\\{{k\\choose d+2i-1} +{k\\choose d+2i-2}{v-k\\choose 1}+\\cdots\\right.$ $\\left.+{k\\choose d+i}{v-k\\choose i-1} \\right\\}$ holds for each $i$ satisfying $1\\leq i\\leq k-d$, 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}\\leq v \\leq \\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.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-03-18T08:40:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E30",
      "05B30"
    ],
    "keywords": [
      "upper bound",
      "lower bound",
      "maximal cardinality",
      "point subsets",
      "point set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.3821B"
    }
  }
}