{
  "id": "1804.00321",
  "version": "v1",
  "published": "2018-04-01T17:27:24.000Z",
  "updated": "2018-04-01T17:27:24.000Z",
  "title": "A magic rectangle set on Abelian groups",
  "authors": [
    "Sylwia Cichacz",
    "Tomasz Hinc"
  ],
  "journal": "Discrete Applied Mathematics 288 (2021) 201-210",
  "doi": "10.1016/j.dam.2020.08.029",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "A $\\Gamma$-magic rectangle set $MRS_{\\Gamma}(a, b; c)$ of order $abc$ is a collection of $c$ arrays $(a\\times b)$ whose entries are elements of group $\\Gamma$, each appearing once, with all row sums in every rectangle equal to a constant $\\omega\\in \\Gamma$ and all column sums in every rectangle equal to a constant $\\delta \\in \\Gamma$. In this paper we prove that for $\\{a,b\\}\\neq\\{2^{\\alpha},2k+1\\}$ where $\\alpha$ and $k$ are some natural numbers, a $\\Gamma$-magic rectangle set MRS$_{\\Gamma}(a, b;c)$ exists if and only if $a$ and $b$ are both even or and $|\\Gamma|$ is odd or $\\Gamma$ has more than one involution. Moreover we obtain sufficient and necessary conditions for existence a $\\Gamma$-magic rectangle MRS$_{\\Gamma}(a, b)$=MRS$_{\\Gamma}(a, b;1)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-01T17:27:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "abelian groups",
      "magic rectangle set mrs",
      "rectangle equal",
      "magic rectangle mrs",
      "row sums"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}