{
  "id": "2204.01545",
  "version": "v1",
  "published": "2022-04-04T14:52:03.000Z",
  "updated": "2022-04-04T14:52:03.000Z",
  "title": "A General Construction of Permutation Polynomials of $\\Bbb F_{q^2}$",
  "authors": [
    "Xiang-dong Hou",
    "Vincenzo Pallozzi Lavorante"
  ],
  "comment": "30 pages, 2 figures",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $r$ be a positive integer, $h(X)\\in\\Bbb F_{q^2}[X]$, and $\\mu_{q+1}$ be the subgroup of order $q+1$ of $\\Bbb F_{q^2}^*$. It is well known that $X^rh(X^{q-1})$ permutes $\\Bbb F_{q^2}$ if and only if $\\text{gcd}(r,q-1)=1$ and $X^rh(X)^{q-1}$ permutes $\\mu_{q+1}$. There are many ad hoc constructions of permutation polynomials of $\\Bbb F_{q^2}$ of this type such that $h(X)^{q-1}$ induces monomial functions on the cosets of a subgroup of $\\mu_{q+1}$. We give a general construction that can generate, through an algorithm, {\\em all} permutation polynomials of $\\Bbb F_{q^2}$ with this property, including many which are not known previously. The construction is illustrated explicitly for permutation binomials and trinomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-04T14:52:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11T06",
      "11T30",
      "11T55"
    ],
    "keywords": [
      "permutation polynomials",
      "general construction",
      "induces monomial functions",
      "ad hoc constructions",
      "permutation binomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}