{
  "id": "1601.08009",
  "version": "v1",
  "published": "2016-01-29T09:11:34.000Z",
  "updated": "2016-01-29T09:11:34.000Z",
  "title": "Classification of k-nets",
  "authors": [
    "G. Korchmáros",
    "G. P. Nagy"
  ],
  "journal": "European Journal of Combinatorics 48: pp. 177-185. (2015)",
  "doi": "10.1016/j.ejc.2015.02.019",
  "categories": [
    "math.CO"
  ],
  "abstract": "A finite \\emph{$k$-net} of order $n$ is an incidence structure consisting of $k\\ge 3$ pairwise disjoint classes of lines, each of size $n$, such that every point incident with two lines from distinct classes is incident with exactly one line from each of the $k$ classes. Deleting a line class from a $k$-net, with $k\\ge 4$, gives a \\emph{derived} ($k-1$)-net of the same order. Finite $k$-nets embedded in a projective plane $PG(2,K)$ coordinatized by a field $K$ of characteristic $0$ only exist for $k=3,4$, see \\cite{knp_k}. In this paper, we investigate $3$-nets embedded in $PG(2,K)$ whose line classes are in perspective position with an axis $r$, that is, every point on the line $r$ incident with a line of the net is incident with exactly one line from each class. The problem of determining all such $3$-nets remains open whereas we obtain a complete classification for those coordinatizable by a group. As a corollary, the (unique) $4$-net of order $3$ embedded in $PG(2,K)$ turns out to be the only $4$-net embedded in $PG(2,K)$ with a derived $3$-net which can be coordinatized by a group. Our results hold true in positive characteristic under the hypothesis that the order of the $k$-net considered is smaller than the characteristic of $K$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-29T09:11:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52C30",
      "05B25"
    ],
    "keywords": [
      "line class",
      "characteristic",
      "nets remains open",
      "results hold true",
      "pairwise disjoint classes"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160108009K"
    }
  }
}