{
  "id": "1910.02753",
  "version": "v1",
  "published": "2019-10-07T12:35:44.000Z",
  "updated": "2019-10-07T12:35:44.000Z",
  "title": "Enumerating extensions of mutually orthogonal Latin squares",
  "authors": [
    "Simona Boyadzhiyska",
    "Shagnik Das",
    "Tibor Szabó"
  ],
  "comment": "18 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Two $n \\times n$ Latin squares $L_1, L_2$ are said to be orthogonal if, for every ordered pair $(x,y)$ of symbols, there are coordinates $(i,j)$ such that $L_1(i,j) = x$ and $L_2(i,j) = y$. A $k$-MOLS is a sequence of $k$ pairwise-orthogonal Latin squares, and the existence and enumeration of these objects has attracted a great deal of attention. Recent work of Keevash and Luria provides, for all fixed $k$, log-asymptotically tight bounds on the number of $k$-MOLS. To study the situation when $k$ grows with $n$, we bound the number of ways a $k$-MOLS can be extended to a $(k+1)$-MOLS. These bounds are again tight for constant $k$, and allow us to deduce upper bounds on the total number of $k$-MOLS for all $k$. These bounds are close to tight even for $k$ linear in $n$, and readily generalize to the broader class of gerechte designs, which include Sudoku squares.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-07T12:35:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B15"
    ],
    "keywords": [
      "mutually orthogonal latin squares",
      "enumerating extensions",
      "deduce upper bounds",
      "pairwise-orthogonal latin squares",
      "broader class"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}