{
  "id": "2202.06237",
  "version": "v1",
  "published": "2022-02-13T07:35:25.000Z",
  "updated": "2022-02-13T07:35:25.000Z",
  "title": "Codes and Designs in Johnson Graphs From Symplectic Actions on Quadratic Forms",
  "authors": [
    "John Bamberg",
    "Alice Devillers",
    "Mark Ioppolo",
    "Cheryl E. Praeger"
  ],
  "categories": [
    "math.CO",
    "math.GR"
  ],
  "abstract": "The Johnson graph $J(v, k)$ has as vertices the $k$-subsets of $\\mathcal{V}=\\{1,\\ldots, v\\}$, and two vertices are joined by an edge if their intersection has size $k-1$. An \\emph{$X$-strongly incidence-transitive code} in $J (v, k)$ is a proper vertex subset $\\Gamma$ such that the subgroup $X$ of graph automorphisms leaving $\\Gamma$ invariant is transitive on the set $\\Gamma$ of `codewords', and for each codeword $\\Delta$, the setwise stabiliser $X_\\Delta$ is transitive on $\\Delta \\times (\\mathcal{V}\\setminus \\Delta)$. We classify the \\emph{$X$-strongly incidence-transitive codes} in $J(v,k)$ for which $X$ is the symplectic group $\\mathrm{Sp}_{2n}(2)$ acting as a $2$-transitive permutation group of degree $2^{2n-1}\\pm 2^{n-1}$, where the stabiliser $X_\\Delta$ of a codeword $\\Delta$ is contained in a \\emph{geometric} maximal subgroup of $X$. In particular, we construct two new infinite families of strongly incidence-transitive codes associated with the reducible maximal subgroups of $\\mathrm{Sp}_{2n}(2)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-13T07:35:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C25",
      "05E18",
      "20B25",
      "94B25"
    ],
    "keywords": [
      "johnson graph",
      "quadratic forms",
      "symplectic actions",
      "strongly incidence-transitive code",
      "proper vertex subset"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}