{
  "id": "1801.07038",
  "version": "v1",
  "published": "2018-01-22T11:02:44.000Z",
  "updated": "2018-01-22T11:02:44.000Z",
  "title": "A coding theoretic approach to the uniqueness conjecture for projective planes of prime order",
  "authors": [
    "Bhaskar Bagchi"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "An outstanding folklore conjecture asserts that, for any prime $p$, up to isomorphism the projective plane $PG(2,\\mathbb{F}_p)$ over the field $\\mathbb{F}_p := \\mathbb{Z}/p\\mathbb{Z}$ is the unique projective plane of order $p$. Let $\\pi$ be any projective plane of order $p$. For any partial linear space ${\\cal X}$, define the inclusion number $i({\\cal X},\\pi)$ to be the number of isomorphic copies of ${\\cal X}$ in $\\pi$. In this paper we prove that if ${\\cal X}$ has at most $\\log_2 p$ lines, then $i({\\cal X},\\pi)$ can be written as an explicit rational linear combination (depending only on ${\\cal X}$ and $p$) of the coefficients of the complete weight enumerator (c.w.e.) of the $p$-ary code of $\\pi$. Thus, the c.w.e. of this code carries an enormous amount of structural information about $\\pi$. In consequence, it is shown that if $p > 2^ 9=512$, and $\\pi$ has the same c.w.e. as $PG(2,\\mathbb{F}_p)$, then $\\pi$ must be isomorphic to $PG(2,\\mathbb{F}_p)$. Thus, the uniqueness conjecture can be approached via a thorough study of the possible c.w.e. of the codes of putative projective planes of prime order.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-22T11:02:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "projective plane",
      "coding theoretic approach",
      "uniqueness conjecture",
      "prime order",
      "explicit rational linear combination"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}