{
  "id": "2406.00243",
  "version": "v1",
  "published": "2024-06-01T00:21:04.000Z",
  "updated": "2024-06-01T00:21:04.000Z",
  "title": "There are no good infinite families of toric codes",
  "authors": [
    "Jason P. Bell",
    "Sean Monahan",
    "Matthew Satriano",
    "Karen Situ",
    "Zheng Xie"
  ],
  "comment": "10 pages. Comments welcome",
  "categories": [
    "math.CO",
    "cs.IT",
    "math.AG",
    "math.IT"
  ],
  "abstract": "Soprunov and Soprunova introduced the notion of a good infinite family of toric codes. We prove that such good families do not exist by proving a more general Szemer\\'edi-type result: for all $c\\in(0,1]$ and all positive integers $N$, subsets of density at least $c$ in $\\{0,1,\\dots,N-1\\}^n$ contain hypercubes of arbitrarily large dimension as $n$ grows.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-01T00:21:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14G50",
      "14M25",
      "11B30",
      "94B05"
    ],
    "keywords": [
      "toric codes",
      "infinite family",
      "general szemeredi-type result",
      "contain hypercubes",
      "arbitrarily large dimension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}