{
  "id": "2006.16352",
  "version": "v1",
  "published": "2020-06-29T20:17:26.000Z",
  "updated": "2020-06-29T20:17:26.000Z",
  "title": "Cameron-Liebler line classes",
  "authors": [
    "Morgan Rodgers"
  ],
  "journal": "Des. Codes Cryptogr. 68 (2013), 33-37",
  "doi": "10.1007/s10623-011-9581-2",
  "categories": [
    "math.CO"
  ],
  "abstract": "New examples of Cameron-Liebler line classes in $\\mathrm{PG}(3,q)$ are given with parameter $\\frac{1}{2}(q^2 -1)$. These examples have been constructed for many odd values of $q$ using a computer search, by forming a union of line orbits from a cyclic collineation group acting on the space. While there are many equivalent characterizations of these objects, perhaps the most significant is that a set of lines $\\mathcal{L}$ in $\\mathrm{PG}(3,q)$ is a Cameron-Liebler line class with parameter $x$ if and only if every spread $\\mathcal{S}$ of the space shares precisely $x$ lines with $\\mathcal{L}$. These objects are related to generalizations of symmetric tactical decompositions of $\\mathrm{PG}(3,q)$, as well as to subgroups of $\\mathrm{P\\Gamma L}(4,q)$ having equally many orbits on points and lines of $\\mathrm{PG}(3,q)$. Furthermore, in some cases the line classes we construct are related to two-intersection sets in $\\mathrm{AG}(2,q)$. Since there are very few known examples of these sets for $q$ odd, any new results in this direction are of particular interest.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-29T20:17:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "51A50",
      "51E20"
    ],
    "keywords": [
      "cameron-liebler line class",
      "computer search",
      "space shares",
      "equivalent characterizations",
      "line orbits"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}