{
  "id": "1211.2942",
  "version": "v2",
  "published": "2012-11-13T10:22:20.000Z",
  "updated": "2013-04-13T18:06:03.000Z",
  "title": "(2^n,2^n,2^n,1)-relative difference sets and their representations",
  "authors": [
    "Yue Zhou"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "We show that every $(2^n,2^n,2^n,1)$-relative difference set $D$ in $\\Z_4^n$ relative to $\\Z_2^n$ can be represented by a polynomial $f(x)\\in \\F_{2^n}[x]$, where $f(x+a)+f(x)+xa$ is a permutation for each nonzero $a$. We call such an $f$ a planar function on $\\F_{2^n}$. The projective plane $\\Pi$ obtained from $D$ in the way of Ganley and Spence \\cite{ganley_relative_1975} is coordinatized, and we obtain necessary and sufficient conditions of $\\Pi$ to be a presemifield plane. We also prove that a function $f$ on $\\F_{2^n}$ with exactly two elements in its image set and $f(0)=0$ is planar, if and only if, $f(x+y)=f(x)+f(y)$ for any $x,y\\in\\F_{2^n}$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-04-13T18:06:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "12K10",
      "51A35",
      "51A40"
    ],
    "keywords": [
      "representations",
      "image set",
      "presemifield plane",
      "sufficient conditions",
      "relative difference set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1211.2942Z"
    }
  }
}