{
  "id": "1611.00529",
  "version": "v1",
  "published": "2016-11-02T10:09:46.000Z",
  "updated": "2016-11-02T10:09:46.000Z",
  "title": "Packing Sets over Finite Fields",
  "authors": [
    "Oliver Roche-Newton",
    "Arne Winterhof"
  ],
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "For a given subset $A\\subseteq \\mathbb F_q^*$, we study the problem of finding a large packing set $B$ of $A$, that is, a set $B \\subseteq \\mathbb F_q^*$ such that $|AB|=|A||B|$. We prove the existence of such a $B$ of size $|B|\\ge (q-1)/|A/A|$ and show that this bound is in general optimal. The case that $q=p$ is a prime and $A=\\{1,2,\\ldots,\\lambda\\}$ for some positive integer $\\lambda$ is particularly interesting in view of the construction of limited-magnitude error correcting codes. Here we construct a packing set $B$ of size $|B|\\gg p (\\lambda \\log p)^{-1}$ for any $\\lambda \\le c p^{1/2}$ for some explicitly calcuable constant $c$. This result is optimal up to the logarithmic factor.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-02T10:09:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B30",
      "11N69"
    ],
    "keywords": [
      "finite fields",
      "limited-magnitude error correcting codes",
      "large packing set",
      "logarithmic factor",
      "general optimal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}