{
  "id": "1705.06372",
  "version": "v1",
  "published": "2017-05-17T23:46:41.000Z",
  "updated": "2017-05-17T23:46:41.000Z",
  "title": "Elation KM-arcs",
  "authors": [
    "Maarten De Boeck",
    "Geertrui Van de Voorde"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper, we study KM-arcs in $PG(2, q)$, the Desarguesian projective plane of order $q$. A KM-arc A of type $t$ is a natural generalisation of a hyperoval: it is a set of $q + t$ points in $PG(2, q)$ such that every line of $PG(2,q)$ meets A in $0,2$ or $t$ points. We study a particular class of KM-arcs, namely, elation KM-arcs. These KM-arcs are highly symmetrical and moreover, many of the known examples are elation KM-arcs. We provide an algebraic framework and show that all elation KM-arcs of type $q/4$ in $PG(2,q)$ are translation KM-arcs. Using a result of [2], this concludes the classification problem for elation KM-arcs of type $q/4$. Furthermore, we construct for all $q = 2^h$, $h > 3$, an infinite family of elation KM-arcs of type $q/8$, and for $q = 2^h$, where $4, 6, 7 | h$ an infinite family of KM-arcs of type $q/16$. Both families contain new examples of KM-arcs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-17T23:46:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "elation km-arcs",
      "desarguesian projective plane",
      "natural generalisation",
      "study km-arcs",
      "algebraic framework"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}