{
  "id": "2105.07979",
  "version": "v2",
  "published": "2021-05-17T16:02:55.000Z",
  "updated": "2021-05-20T15:07:43.000Z",
  "title": "Designs, permutations, and transitive groups",
  "authors": [
    "Minjia Shi",
    "XiaoXiao Li",
    "Patrick Solé"
  ],
  "comment": "12 pages. arXiv admin note: text overlap with arXiv:2102.08276",
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "A notion of $t$-designs in the symmetric group on $n$ letters was introduced by Godsil in 1988. In particular $t$-transitive sets of permutations form a $t$-design. We derive special lower bounds for $t=1$ and $t=2$ by a power moment method. For general $n,t$ we give a %linear programming lower bound . For $n\\ge 4$ and $t=2,$ this bound is strong enough to show a lower bound on the size of such $t$-designs of $n(n-1)\\dots (n-t+1),$ which is best possible when sharply $t$-transitive sets of permutations exist. This shows, in particular, that tight $2$-designs do not exist.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2021-05-20T15:07:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E35",
      "05E24"
    ],
    "keywords": [
      "transitive groups",
      "power moment method",
      "derive special lower bounds",
      "transitive sets",
      "programming lower bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}