{
  "id": "2009.03475",
  "version": "v1",
  "published": "2020-09-08T01:34:05.000Z",
  "updated": "2020-09-08T01:34:05.000Z",
  "title": "Maximal sets of mutually orthogonal frequency squares",
  "authors": [
    "Nicholas J. Cavenagh",
    "Adam Mammoliti",
    "Ian M. Wanless"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A frequency square is a square matrix in which each row and column is a permutation of the same multiset of symbols. A frequency square is of type $(n;\\lambda)$ if it contains $n/\\lambda$ symbols, each of which occurs $\\lambda$ times per row and $\\lambda$ times per column. In the case when $\\lambda=n/2$ we refer to the frequency square as binary. A set of $k$-MOFS$(n;\\lambda)$ is a set of $k$ frequency squares of type $(n;\\lambda)$ such that when any two of the frequency squares are superimposed, each possible ordered pair occurs equally often. A set of $k$-maxMOFS$(n;\\lambda)$ is a set of $k$-MOFS$(n;\\lambda)$ that is not contained in any set of $(k+1)$-MOFS$(n;\\lambda)$. For even $n$, let $\\mu(n)$ be the smallest $k$ such that there exists a set of $k$-maxMOFS$(n;n/2)$. It was shown in [Electron. J. Combin. 27(3) (2020), P3.7] that $\\mu(n)=1$ if $n/2$ is odd and $\\mu(n)>1$ if $n/2$ is even. Extending this result, we show that if $n/2$ is even, then $\\mu(n)>2$. Also, we show that whenever $n$ is divisible by a particular function of $k$, there does not exist a set of $k'$-maxMOFS$(n;n/2)$ for any $k'\\le k$. In particular, this means that $\\limsup \\mu(n)$ is unbounded. Nevertheless we can construct infinite families of maximal binary MOFS of fixed cardinality. More generally, let $q=p^u$ be a prime power and let $p^v$ be the highest power of $p$ that divides $n$. If $0\\le v-uh<u/2$ for $h\\ge1$ then we show that there exists a set of $(q^h-1)^2/(q-1)$-maxMOFS$(n;n/q)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-08T01:34:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B15"
    ],
    "keywords": [
      "mutually orthogonal frequency squares",
      "maximal sets",
      "maximal binary mofs",
      "construct infinite families",
      "prime power"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}