{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "M9XF82GGW8"
    },
    {
      "field": "uses",
      "target": "QHPW53E7YA"
    },
    {
      "field": "uses",
      "target": "R43K0GTTDS"
    }
  ],
  "body": "\n## Definition\n\nLet $t, v, k, \\lambda$ be positive integers with $t \\le k \u003c v$.\nA $t$-$(v, k, \\lambda)$ design is a pair $(X, \\mathcal{B})$ such that:\n\n1. $X$ is a [[finite set|0BBTH3YSZA]] with $v$ elements, called points.\n2. $\\mathcal{B}$ is a finite [[family|05Q6G4EH2G]] of subsets of $X$, called blocks, each with exactly $k$ elements.\n3. Each subset of $X$ with exactly $t$ elements is contained in exactly $\\lambda$ blocks.\n\nA balanced incomplete block design, or $2$-design, is a $2$-$(v, k, \\lambda)$ design.\n\n## Remarks\n\nThe family $\\mathcal{B}$ may contain a block more than once.\nThe design is simple if it has no repeated block.\nIn a $2$-$(v, k, \\lambda)$ design with $b$ blocks, each point lies in the same number $r$ of blocks, and $bk = vr$ and $r(k - 1) = \\lambda (v - 1)$.\nExample: let $X = \\set{1, \\dots, 7}$ and let the blocks be $\\set{1,2,3}$, $\\set{1,4,5}$, $\\set{1,6,7}$, $\\set{2,4,6}$, $\\set{2,5,7}$, $\\set{3,4,7}$, $\\set{3,5,6}$.\nEach of the $21$ pairs of points lies in exactly one of these blocks, so this is a $2$-$(7, 3, 1)$ design, the [[Fano plane|QQ8GW1ASZM]].\nHere each point lies in $r = 3$ blocks, and $b = 7$.\nNon-example: on $X = \\set{1, 2, 3, 4}$ the blocks $\\set{1,2,3}$ and $\\set{1,2,4}$ do not form a $2$-design, because the pair $\\set{1,2}$ lies in two blocks and the pair $\\set{3,4}$ lies in none.\nA $t$-$(v, k, 1)$ design is a [[Steiner system|7BPSC4T4M3]] $S(t, k, v)$.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "4JCSS2Z1BA",
    "kind": "definition",
    "msc": [
      "05B05"
    ],
    "name": "Block design",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Block_design"
      }
    ],
    "tag": "TGZATGBHTH",
    "uses": [
      "0BBTH3YSZA",
      "05Q6G4EH2G"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/TGZATGBHTH/",
  "links": [
    {
      "field": "defines",
      "target": "4JCSS2Z1BA"
    },
    {
      "field": "uses",
      "target": "0BBTH3YSZA"
    },
    {
      "field": "uses",
      "target": "05Q6G4EH2G"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/TGZATGBHTH.md",
  "msc": [
    "05B05"
  ],
  "name": "Block design",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/TGZ/TGZATGBHTH.md",
  "status": null,
  "tag": "TGZATGBHTH",
  "uri": "m2://TGZATGBHTH",
  "url": "https://math2.tamnd.com/wiki/block-design-definition/",
  "writing": {
    "tag": "TGZATGBHTH",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}