{
  "id": "2411.19201",
  "version": "v1",
  "published": "2024-11-28T15:16:32.000Z",
  "updated": "2024-11-28T15:16:32.000Z",
  "title": "Multisets with few special directions and small weight codewords in Desarguesian planes",
  "authors": [
    "Sam Adriaensen",
    "Tamás Szőnyi",
    "Zsuzsa Weiner"
  ],
  "comment": "25 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper, we tie together two well studied topics related to finite Desarguesian affine and projective planes. The first topic concerns directions determined by a set, or even a multiset, of points in an affine plane. The second topic concerns the linear code generated by the incidence matrix of a projective plane. We show how a multiset determining only $k$ special directions, in a modular sense, gives rise to a codeword whose support can be covered by $k$ concurrent lines. The reverse operation of going from a codeword to a multiset of points is trickier, but we describe a possible strategy and show some fruitful applications. Given a multiset of affine points, we use a bound on the degree of its so-called projection function to yield lower bounds on the number of special directions, both in an ordinary and in a modular sense. In the codes related to projective planes of prime order $p$, there exists an odd codeword, whose support is covered by 3 concurrent lines, but which is not a linear combination of these 3 lines. We generalise this codeword to codewords whose support is contained in an arbitrary number of concurrent lines. In case $p$ is large enough, this allows us to extend the classification of codewords from weight at most $4p-22$ to weight at most $5p-36$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-28T15:16:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B25",
      "94B05"
    ],
    "keywords": [
      "small weight codewords",
      "special directions",
      "desarguesian planes",
      "concurrent lines",
      "projective plane"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}