{
  "id": "2003.10337",
  "version": "v1",
  "published": "2020-03-23T15:48:33.000Z",
  "updated": "2020-03-23T15:48:33.000Z",
  "title": "Small Weight Code Words of Projective Geometric Codes",
  "authors": [
    "Sam Adriaensen",
    "Lins Denaux"
  ],
  "comment": "26 pages, 1 figure",
  "categories": [
    "math.CO"
  ],
  "abstract": "We investigate small weight code words of the $p$-ary linear code $\\mathcal C_{j,k}(n,q)$ generated by the incidence matrix of $k$-spaces and $j$-spaces of PG$(n,q)$ and its dual, with $q$ a prime power and $0 \\leq j < k < n$. Firstly, we prove that all code words of $\\mathcal C_{j,k}(n,q)$ up to weight $\\left(3 - \\mathcal{O}\\left(\\frac 1 q \\right) \\right) \\genfrac{[}{]}{0pt}{}{k+1}{j+1}_q$ are linear combinations of at most two $k$-spaces (i.e. two rows of the incidence matrix). As for the dual code $\\mathcal C_{j,k}(n,q)^\\perp$, we manage to reduce both problems of determining its minimum weight (1) and characterising its minimum weight code words (2) to the case $\\mathcal C_{0,1}(n,q)^\\perp$. This implies the solution to both problem (1) and (2) if $q$ is prime and the solution to problem (1) if $q$ is even.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-23T15:48:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B25",
      "94B05"
    ],
    "keywords": [
      "small weight code words",
      "projective geometric codes",
      "minimum weight code words",
      "incidence matrix",
      "ary linear code"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}