{
  "id": "1008.5383",
  "version": "v1",
  "published": "2010-08-31T18:40:18.000Z",
  "updated": "2010-08-31T18:40:18.000Z",
  "title": "Bounds on $s$-distance sets with strength $t$",
  "authors": [
    "Hiroshi Nozaki",
    "Sho Suda"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A finite set $X$ in the Euclidean unit sphere is called an $s$-distance set if the set of distances between any distinct two elements of $X$ has size $s$. We say that $t$ is the strength of $X$ if $X$ is a spherical $t$-design but not a spherical $(t+1)$-design. Delsarte-Goethals-Seidel gave an absolute bound for the cardinality of an $s$-distance set. The results of Neumaier and Cameron-Goethals-Seidel imply that if $X$ is a spherical 2-distance set with strength 2, then the known absolute bound for 2-distance sets is improved. This bound are also regarded as that for a strongly regular graph with the certain condition of the Krein parameters. In this paper, we give two generalizations of this bound to spherical $s$-distance sets with strength $t$ (more generally, to $s$-distance sets with strength $t$ in a two-point-homogeneous space), and to $Q$-polynomial association schemes. First, for any $s$ and $s-1 \\leq t \\leq 2s-2$, we improve the known absolute bound for the size of a spherical $s$-distance set with strength $t$. Secondly, for any $d$, we give an absolute bound for the size of a $Q$-polynomial association scheme of class $d$ with the certain conditions of the Krein parameters.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-08-31T18:40:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "distance set",
      "absolute bound",
      "polynomial association scheme",
      "krein parameters",
      "euclidean unit sphere"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1008.5383N"
    }
  }
}