{
  "id": "2007.00911",
  "version": "v1",
  "published": "2020-07-02T06:36:47.000Z",
  "updated": "2020-07-02T06:36:47.000Z",
  "title": "Algebraic constructions of complete $m$-arcs",
  "authors": [
    "Daniele Bartoli",
    "Giacomo Micheli"
  ],
  "categories": [
    "math.CO",
    "math.AG"
  ],
  "abstract": "Let $m$ be a positive integer, $q$ be a prime power, and $\\mathrm{PG}(2,q)$ be the projective plane over the finite field $\\mathbb F_q$. Finding complete $m$-arcs in $\\mathrm{PG}(2,q)$ of size less than $q$ is a classical problem in finite geometry. In this paper we give a complete answer to this problem when $q$ is relatively large compared with $m$, explicitly constructing the smallest $m$-arcs in the literature so far for any $m\\geq 8$. For any fixed $m$, our arcs $\\mathcal A_{q,m}$ satisfy $|\\mathcal A_{q,m}|-q\\rightarrow -\\infty$ as $q$ grows. To produce such $m$-arcs, we develop a Galois theoretical machinery that allows the transfer of geometric information of points external to the arc, to arithmetic one, which in turn allows to prove the $m$-completeness of the arc.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-02T06:36:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "algebraic constructions",
      "finite field",
      "complete answer",
      "prime power",
      "galois theoretical machinery"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}