{
  "id": "2304.06220",
  "version": "v1",
  "published": "2023-04-13T02:26:09.000Z",
  "updated": "2023-04-13T02:26:09.000Z",
  "title": "Jacobi polynomials and design theory II",
  "authors": [
    "Himadri Chakraborty",
    "Reina Ishikawa",
    "Yuuho Tanaka"
  ],
  "comment": "28 pages",
  "categories": [
    "math.CO",
    "math.GR",
    "math.NT"
  ],
  "abstract": "In this paper, we introduce some new polynomials associated to linear codes over $\\mathbb{F}_{q}$. In particular, we introduce the notion of split complete Jacobi polynomials attached to multiple sets of coordinate places of a linear code over $\\mathbb{F}_{q}$, and give the MacWilliams type identity for it. We also give the notion of generalized $q$-colored $t$-designs. As an application of the generalized $q$-colored $t$-designs, we derive a formula that obtains the split complete Jacobi polynomials of a linear code over $\\mathbb{F}_{q}$.Moreover, we define the concept of colored packing (resp. covering) designs. Finally, we give some coding theoretical applications of the colored designs for Type~III and Type~IV codes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-13T02:26:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11T71",
      "94B05",
      "11F11"
    ],
    "keywords": [
      "design theory",
      "split complete jacobi polynomials",
      "linear code",
      "macwilliams type identity",
      "coordinate places"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}