{
  "id": "1703.04764",
  "version": "v1",
  "published": "2017-03-14T22:15:14.000Z",
  "updated": "2017-03-14T22:15:14.000Z",
  "title": "Parity of Sets of Mutually Orthogonal Latin Squares",
  "authors": [
    "Nevena Francetić",
    "Sarada Herke",
    "Ian M. Wanless"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Every Latin square has three attributes that can be even or odd, but any two of these attributes determines the third. Hence the parity of a Latin square has an information content of 2 bits. We extend the definition of parity from Latin squares to sets of mutually orthogonal Latin squares (MOLS) and the corresponding orthogonal arrays (OA). Suppose the parity of an $\\mathrm{OA}(k,n)$ has an information content of $\\dim(k,n)$ bits. We show that $\\dim(k,n) \\leq {k \\choose 2}-1$. For the case corresponding to projective planes we prove a tighter bound, namely $\\dim(n+1,n) \\leq {n \\choose 2}$ when $n$ is odd and $\\dim(n+1,n) \\leq {n \\choose 2}-1$ when $n$ is even. Using the existence of MOLS with subMOLS, we prove that if $\\dim(k,n)={k \\choose 2}-1$ then $\\dim(k,N) = {k \\choose 2}-1$ for all sufficiently large $N$. Let the ensemble of an $\\mathrm{OA}$ be the set of Latin squares derived by interpreting any three columns of the OA as a Latin square. We demonstrate many restrictions on the number of Latin squares of each parity that the ensemble of an $\\mathrm{OA}(k,n)$ can contain. These restrictions depend on $n\\mod4$ and give some insight as to why it is harder to build projective planes of order $n \\not= 2\\mod4$ than for $n \\not= 2\\mod4$. For example, we prove that when $n \\not= 2\\mod 4$ it is impossible to build an $\\mathrm{OA}(n+1,n)$ for which all Latin squares in the ensemble are isotopic (equivalent to each other up to permutation of the rows, columns and symbols).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-14T22:15:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B15",
      "05B25"
    ],
    "keywords": [
      "mutually orthogonal latin squares",
      "information content",
      "build projective planes",
      "corresponding orthogonal arrays",
      "attributes determines"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}