{
  "id": "2212.03694",
  "version": "v1",
  "published": "2022-12-07T15:10:47.000Z",
  "updated": "2022-12-07T15:10:47.000Z",
  "title": "An upper bound on the number of frequency hypercubes",
  "authors": [
    "Denis S. Krotov",
    "Vladimir N. Potapov"
  ],
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "A frequency $n$-cube $F^n(q;l_0,...,l_{m-1})$ is an $n$-dimensional $q$-by-...-by-$q$ array, where $q = l_0+...+l_{m-1}$, filled by numbers $0,...,m-1$ with the property that each line contains exactly $l_i$ cells with symbol $i$, $i = 0,...,m-1$ (a line consists of $q$ cells of the array differing in one coordinate). The trivial upper bound on the number of frequency $n$-cubes is $m^{(q-1)^{n}}$. We improve that lower bound for $n>2$, replacing $q-1$ by a smaller value, by constructing a testing set of size $s^{n}$, $s<q-1$, for frequency $n$-cubes (a testing sets is a collection of cells of an array the values in which uniquely determine the array with given parameters). We also construct new testing sets for generalized frequency $n$-cubes, which are essentially correlation immune functions in $n$ $q$-valued arguments; the cardinalities of new testing sets are smaller than for testing sets known before.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-07T15:10:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B15"
    ],
    "keywords": [
      "testing set",
      "frequency hypercubes",
      "trivial upper bound",
      "essentially correlation immune functions",
      "line consists"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}