{
  "id": "2005.10887",
  "version": "v1",
  "published": "2020-05-21T20:22:17.000Z",
  "updated": "2020-05-21T20:22:17.000Z",
  "title": "On the number of frequency hypercubes $F^n(4;2,2)$",
  "authors": [
    "Minjia Shi",
    "Shukai Wang",
    "Xiaoxiao Li",
    "Denis S. Krotov"
  ],
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "A frequency $n$-cube $F^n(4;2,2)$ is an $n$-dimensional $4\\times\\cdots\\times 4$ array filled by $0$s and $1$s such that each line contains exactly two $1$s. We classify the frequency $4$-cubes $F^4(4;2,2)$, find a testing set of size $25$ for $F^3(4;2,2)$, and derive an upper bound on the number of $F^n(4;2,2)$. Additionally, for any $n$ greater than $2$, we construct an $F^n(4;2,2)$ that cannot be refined to a latin hypercube, while each of its sub-$F^{n-1}(4;2,2)$ can. Keywords: frequency hypercube, frequency square, latin hypercube, testing set, MDS code",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-21T20:22:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B15"
    ],
    "keywords": [
      "frequency hypercube",
      "latin hypercube",
      "testing set",
      "line contains",
      "mds code"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}