{
  "id": "2309.00490",
  "version": "v1",
  "published": "2023-09-01T14:33:29.000Z",
  "updated": "2023-09-01T14:33:29.000Z",
  "title": "Small weight codewords of projective geometric codes II",
  "authors": [
    "Sam Adriaensen",
    "Lins Denaux"
  ],
  "comment": "22 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "The $p$-ary linear code $\\mathcal C_{k}(n,q)$ is defined as the row space of the incidence matrix $A$ of $k$-spaces and points of $\\text{PG}(n,q)$. It is known that if $q$ is square, a codeword of weight $q^k\\sqrt{q}+\\mathcal O \\left( q^{k-1} \\right) $ exists that cannot be written as a linear combination of at most $\\sqrt{q}$ rows of $A$. Over the past few decades, researchers have put a lot of effort towards proving that any codeword of smaller weight does meet this property. We show that if $ q \\geqslant 32 $ is a composite prime power, every codeword of $\\mathcal C_k(n,q)$ up to weight $\\mathcal O \\left( {q^k\\sqrt{q}} \\right) $ is a linear combination of at most $\\sqrt{q}$ rows of $A$. We also generalise this result to the codes $\\mathcal C_{j,k}(n,q) $, which are defined as the $p$-ary row span of the incidence matrix of $k$-spaces and $j$-spaces, $j < k$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-01T14:33:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B25",
      "94B05"
    ],
    "keywords": [
      "small weight codewords",
      "projective geometric codes",
      "linear combination",
      "incidence matrix",
      "ary linear code"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}