{
  "id": "1903.10309",
  "version": "v1",
  "published": "2019-03-25T13:35:52.000Z",
  "updated": "2019-03-25T13:35:52.000Z",
  "title": "Permutation polynomials of degree 8 over finite fields of characteristic 2",
  "authors": [
    "Xiang Fan"
  ],
  "comment": "16 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Up to linear transformations, we obtain a classification of permutation polynomials (PPs) of degree $8$ over $\\mathbb{F}_{2^r}$ with $r>3$. By [J. Number Theory 176 (2017) 466-66], a polynomial $f$ of degree $8$ over $\\mathbb{F}_{2^r}$ is exceptional if and only if $f-f(0)$ is a linearized PP. So it suffices to search for non-exceptional PPs of degree $8$ over $\\mathbb{F}_{2^r}$, which exist only when $r\\leqslant9$ by a previous result. This can be exhausted by the SageMath software running on a personal computer. To facilitate the computation, some requirements after linear transformations and explicit equations by Hermite's criterion are provided for the polynomial coefficients. The main result is that a non-exceptional PP $f$ of degree $8$ over $\\mathbb{F}_{2^r}$ (with $r>3$) exists if and only if $r\\in\\{4,5,6\\}$, and such $f$ is explicitly listed up to linear transformations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-25T13:35:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11T06",
      "12Y05"
    ],
    "keywords": [
      "permutation polynomials",
      "finite fields",
      "linear transformations",
      "characteristic",
      "non-exceptional pp"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}