{
  "id": "2305.11955",
  "version": "v1",
  "published": "2023-05-19T18:32:24.000Z",
  "updated": "2023-05-19T18:32:24.000Z",
  "title": "New bounds for covering codes of radius 3 and codimension 3t+1",
  "authors": [
    "Alexander A. Davydov",
    "Stefano Marcugini",
    "Fernanda Pambianco"
  ],
  "comment": "19 pages, 3 figures",
  "categories": [
    "math.CO",
    "cs.IT",
    "math.IT"
  ],
  "abstract": "The smallest possible length of a $q$-ary linear code of covering radius $R$ and codimension (redundancy) $r$ is called the length function and is denoted by $\\ell_q(r,R)$. In this work, for $q$ \\emph{an arbitrary prime power}, we obtain the following new constructive upper bounds on $\\ell_q(3t+1,3)$: $\\ell_q(r,3)\\lessapprox \\sqrt[3]{k}\\cdot q^{(r-3)/3}\\cdot\\sqrt[3]{\\ln q},~r=3t+1, ~t\\ge1, ~ q\\ge\\lceil\\mathcal{W}(k)\\rceil, 18 <k\\le20.339,~\\mathcal{W}(k)\\text{ is a decreasing function of }k ;$ $\\ell_q(r,3)\\lessapprox \\sqrt[3]{18}\\cdot q^{(r-3)/3}\\cdot\\sqrt[3]{\\ln q},~r=3t+1,~t\\ge1,~ q\\text{ large enough}.$ For $t = 1$, we use a one-to-one correspondence between codes of covering radius 3 and codimension 4, and 2-saturating sets in the projective space $\\mathrm{PG}(3,q)$. A new construction providing sets of small size is proposed. The codes, obtained by geometrical methods, are taken as the starting ones in the lift-constructions (so-called ``$q^m$-concatenating constructions'') to obtain infinite families of codes with radius 3 and growing codimension $r = 3t + 1$, $t\\ge1$. The new bounds are essentially better than the known ones.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-19T18:32:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "94B05",
      "51E21",
      "51E22"
    ],
    "keywords": [
      "codimension",
      "covering codes",
      "ary linear code",
      "covering radius",
      "arbitrary prime power"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}