{
  "id": "2210.12884",
  "version": "v1",
  "published": "2022-10-23T23:27:42.000Z",
  "updated": "2022-10-23T23:27:42.000Z",
  "title": "Computing the minimum distance of the $C(\\mathbb{O}_{3,6})$ polar Orthogonal Grassmann code with elementary methods",
  "authors": [
    "Sarah Gregory",
    "Fernando Piñero-González",
    "Doel Rivera-Laboy",
    "Lani Southern"
  ],
  "categories": [
    "math.CO",
    "cs.IT",
    "math.AG",
    "math.IT"
  ],
  "abstract": "The polar orthogonal Grassmann code $C(\\mathbb{O}_{3,6})$ is the linear code associated to the Grassmann embedding of the Dual Polar space of $Q^+(5,q)$. In this manuscript we study the minimum distance of this embedding. We prove that the minimum distance of the polar orthogonal Grassmann code $C(\\mathbb{O}_{3,6})$ is $q^3-q^3$ for $q$ odd and $q^3$ for $q$ even. Our technique is based on partitioning the orthogonal space into different sets such that on each partition the code $C(\\mathbb{O}_{3,6})$ is identified with evaluations of determinants of skew--symmetric matrices. Our bounds come from elementary algebraic methods counting the zeroes of particular classes of polynomials. We expect our techniques may be applied to other polar Grassmann codes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-23T23:27:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "polar orthogonal grassmann code",
      "minimum distance",
      "elementary methods",
      "dual polar space",
      "polar grassmann codes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}