{
  "id": "1909.04199",
  "version": "v1",
  "published": "2019-09-09T23:59:29.000Z",
  "updated": "2019-09-09T23:59:29.000Z",
  "title": "Further Results on the Pseudo-$L_{g}(s)$ Association Scheme with $g\\geq 3$, $s\\geq g+2$",
  "authors": [
    "Congwei Wang",
    "Shanqi Pang",
    "Guangzhou Chen"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "It is inevitable that the $L_{g}(s)$ association scheme with $g\\geq 3, s\\geq g+2$ is a pseudo-$L_{g}(s)$ association scheme. On the contrary, although $s^2$ treatments of the pseudo-$L_{g}(s)$ association scheme can form one $L_{g}(s)$ association scheme, it is not always an $L_{g}(s)$ association scheme. Mainly because the set of cardinality $s$, which contains two first-associates treatments of the pseudo-$L_{g}(s)$ association scheme, is non-unique. Whether the order $s$ of a Latin square $\\mathbf{L}$ is a prime power or not, the paper proposes two new conditions in order to extend a $POL(s,w)$ containing $\\mathbf{L}$. It has been known that a $POL(s,w)$ can be extended to a $POL(s,s-1)$ so long as Bruck's \\cite{brh} condition $s\\geq \\frac{(s-1-w)^4-2(s-1-w)^3+2(s-1-w)^2+(s-1-w)}{2}$ is satisfied, Bruck's condition will be completely improved through utilizing six properties of the $L_{w+2}(s)$ association scheme in this paper. Several examples are given to elucidate the application of our results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-09T23:59:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B15"
    ],
    "keywords": [
      "association scheme",
      "latin square",
      "prime power",
      "brucks condition",
      "first-associates treatments"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}