{
  "id": "1609.05657",
  "version": "v1",
  "published": "2016-09-19T10:29:53.000Z",
  "updated": "2016-09-19T10:29:53.000Z",
  "title": "On the smallest size of an almost complete subset of a conic in $\\mathrm{PG}(2,q)$ and extendability of Reed-Solomon codes",
  "authors": [
    "Daniele Bartoli",
    "Alexander A. Davydov",
    "Stefano Marcugini",
    "Fernanda Pambianco"
  ],
  "comment": "20 pages, 20 references, 2 figures, 2 tables",
  "categories": [
    "math.CO"
  ],
  "abstract": "A subset $\\mathcal{S}$ of a conic $\\mathcal{C}$ in the projective plane $\\mathrm{PG}(2,q)$ is called almost complete (AC-subset for short) if it can be extended to a larger arc in $\\mathrm{PG}(2,q)$ only by the points of $\\mathcal{C}\\setminus\\mathcal{S}$ and by the nucleus of $\\mathcal{C}$ when $q$ is even. New upper bounds on the smallest size $t(q)$ of an AC-subset are obtained, in particular \\begin{align*} & t(q)<\\sqrt{q(3\\ln q+\\ln\\ln q +\\ln3)}+\\sqrt{\\frac{q}{3\\ln q}}+4\\thicksim \\sqrt{3}\\sqrt{q\\ln q};\\\\ & t(q)<1.835\\sqrt{q\\ln q}. \\end{align*} The new bounds are used to increase regions of pairs $(N,q)$ for which it is proved that every normal rational curve in $\\mathrm{PG}(N,q)$ is a complete $(q+1)$-arc or, equivalently, that a $[q+1,N+1,q-N+1]_q$ generalized doubly-extended Reed-Solomon code cannot be extended to a $[q+2,N+2,q-N+1]_q$ code.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-19T10:29:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "51E21",
      "51E22",
      "94B05"
    ],
    "keywords": [
      "complete subset",
      "smallest size",
      "extendability",
      "normal rational curve",
      "increase regions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}