{
  "id": "2302.04718",
  "version": "v1",
  "published": "2023-02-09T16:01:56.000Z",
  "updated": "2023-02-09T16:01:56.000Z",
  "title": "A note on small weight codewords of projective geometric codes and on the smallest sets of even type",
  "authors": [
    "Sam Adriaensen"
  ],
  "comment": "14 pages",
  "categories": [
    "math.CO",
    "cs.IT",
    "math.IT"
  ],
  "abstract": "In this paper, we study the codes $\\mathcal C_k(n,q)$ arising from the incidence of points and $k$-spaces in $\\text{PG}(n,q)$ over the field $\\mathbb F_p$, with $q = p^h$, $p$ prime. We classify all codewords of minimum weight of the dual code $\\mathcal C_k(n,q)^\\perp$ in case $q \\in \\{4,8\\}$. This is equivalent to classifying the smallest sets of even type in $\\text{PG}(n,q)$ for $q \\in \\{4,8\\}$. We also provide shorter proofs for some already known results, namely of the best known lower bound on the minimum weight of $\\mathcal C_k(n,q)^\\perp$ for general values of $q$, and of the classification of all codewords of $\\mathcal C_{n-1}(n,q)$ of weight up to $2q^{n-1}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-09T16:01:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "51E20",
      "05B25",
      "94B05"
    ],
    "keywords": [
      "small weight codewords",
      "projective geometric codes",
      "smallest sets",
      "minimum weight",
      "dual code"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}