{
  "id": "2404.00927",
  "version": "v2",
  "published": "2024-04-01T05:08:40.000Z",
  "updated": "2024-09-13T10:33:24.000Z",
  "title": "Construction of Permutation Polynomials over Finite Fields with the help of SCR polynomials",
  "authors": [
    "Bidushi Sharma",
    "Dhiren Kumar Basnet"
  ],
  "comment": "13 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper we take a deeper look at the self conjugate reciprocal (SCR) polynomials, which towards the end of the paper aid the construction of new classes of permutation polynomials of simpler forms over $\\mathbb{F}_{q^{2}}$. The paper focuses on the conditions required for a certain class of degree 2 and degree 3 SCR polynomials to have no roots in $\\mu_{q+1}$ (the set of $(q+1)-\\emph{th}$ roots of unity), which helps in the determination of polynomials that permute $\\mathbb{F}_{q^{2}}$. In the due course we also look upon some higher degree SCR polynomials which can be reduced down to a degree 2 SCR polynomial over both odd and even ordered fields. We further look upon the SCR polynomials of type $ax^{q+1}+bx^{q}+bx+a^{q}$ taking both the cases under consideration viz. $a\\in \\mathbb{F}_{q}$ and $a\\in\\mathbb{F}_{q^{2}}\\setminus\\mathbb{F}_{q}$ both.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-09-13T10:33:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11T06",
      "11T55"
    ],
    "keywords": [
      "permutation polynomials",
      "finite fields",
      "construction",
      "higher degree scr polynomials",
      "self conjugate reciprocal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}