{
  "id": "1511.04937",
  "version": "v1",
  "published": "2015-11-16T12:48:20.000Z",
  "updated": "2015-11-16T12:48:20.000Z",
  "title": "$L_2$ discrepancy of symmetrized generalized Hammersley point sets in base $b$",
  "authors": [
    "Ralph Kritzinger",
    "Lisa M. Kritzinger"
  ],
  "comment": "22 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Two popular and often applied methods to obtain two-dimensional point sets with the optimal order of $L_p$ discrepancy are digit scrambling and symmetrization. In this paper we combine these two techniques and symmetrize $b$-adic Hammersley point sets scrambled with arbitrary permutations. It is already known that these modifications indeed assure that the $L_p$ discrepancy is of optimal order $\\mathcal{O}\\left(\\sqrt{\\log{N}}/N\\right)$ for $p\\in [1,\\infty)$ in contrast to the classical Hammersley point set. We prove an exact formula for the $L_2$ discrepancy of these point sets for special permutations. We also present the permutations which lead to the lowest $L_2$ discrepancy for every base $b\\in\\{2,\\dots,27\\}$ by employing computer search algorithms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-16T12:48:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K06",
      "11K38"
    ],
    "keywords": [
      "symmetrized generalized hammersley point sets",
      "discrepancy",
      "adic hammersley point sets",
      "optimal order",
      "two-dimensional point sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151104937K"
    }
  }
}