{
  "id": "1311.0113",
  "version": "v1",
  "published": "2013-11-01T08:10:34.000Z",
  "updated": "2013-11-01T08:10:34.000Z",
  "title": "Neighbour-transitive codes in Johnson graphs",
  "authors": [
    "Robert A. Liebler",
    "Cheryl E. Praeger"
  ],
  "comment": "30 pages",
  "categories": [
    "math.CO",
    "math.GR"
  ],
  "abstract": "The Johnson graph J(v,k) has, as vertices, the k-subsets of a v-set V, and as edges the pairs of k-subsets with intersection of size k-1. We introduce the notion of a neighbour-transitive code in J(v,k). This is a vertex subset \\Gamma such that the subgroup G of graph automorphisms leaving \\Gamma invariant is transitive on both the set \\Gamma of `codewords' and also the set of `neighbours' of \\Gamma, which are the non-codewords joined by an edge to some codeword. We classify all examples where the group G is a subgroup of the symmetric group on V and is intransitive or imprimitive on the underlying v-set V. In the remaining case where G lies in Sym(V) and G is primitive on V, we prove that, provided distinct codewords are at distance at least 3 in J(v,k), then G is 2-transitive on V. We examine many of the infinite families of finite 2-transitive permutation groups and construct surprisingly rich families of examples of neighbour-transitive codes. A major unresolved case remains.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-11-01T08:10:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C25",
      "20B25",
      "94B60"
    ],
    "keywords": [
      "neighbour-transitive code",
      "johnson graph",
      "major unresolved case remains",
      "construct surprisingly rich families",
      "vertex subset"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.0113L"
    }
  }
}