{
  "id": "1906.00620",
  "version": "v1",
  "published": "2019-06-03T08:07:08.000Z",
  "updated": "2019-06-03T08:07:08.000Z",
  "title": "Sufficient conditions for STS$(3^k)$ of 3-rank $\\leq 3^k-r$ to be resolvable",
  "authors": [
    "Yaqi Lu",
    "Minjia Shi"
  ],
  "comment": "Submitted on 2018",
  "categories": [
    "math.CO"
  ],
  "abstract": "Based on the structure of non-full-$3$-rank $STS(3^k)$ and the orthogonal Latin squares, we mainly give sufficient conditions for $STS(3^k)$ of $3$-rank $\\leq 3^k-r$ to be resolvable in the present paper. Under the conditions, the block set of $STS(3^k)$ can be partitioned into $\\frac{3^k-1}{2}$ parallel classes, i.e., $\\frac{3^k-1}{2}$ $1$-$(v,3,1)$ designs. Finally, we prove that $STS(3^k)$ of 3-rank $\\leq 3^k-r$ is resolvable under the sufficient conditions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-03T08:07:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sufficient conditions",
      "resolvable",
      "orthogonal latin squares",
      "parallel classes",
      "block set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}