{
  "id": "1407.4827",
  "version": "v1",
  "published": "2014-07-14T20:07:33.000Z",
  "updated": "2014-07-14T20:07:33.000Z",
  "title": "Construction of self-dual codes over $\\mathbb{Z}_{2^m}$",
  "authors": [
    "Anuradha Sharma",
    "Amit K. Sharma"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Self-dual codes (Type I and Type II codes) play an important role in the construction of even unimodular lattices, and hence in the determination of Jacobi forms. In this paper, we construct both Type I and Type II codes (of higher lengths) over the ring $\\mathbb{Z}_{2^m}$ of integers modulo $2^m$ from shadows of Type I codes of length $n$ over $\\mathbb{Z}_{2^m}$ for each positive integer $n;$ and obtain their complete weight enumerators. Using these results, we also determine some Jacobi forms on the modular group $\\Gamma(1) = SL(2; \\mathbb{Z}).$ Besides this, for each positive integer $n$; we also construct self-dual codes (of higher lengths) over $\\mathbb{Z}_{2^m}$ from the generalized shadow of a self-dual code $\\mathcal{C}$ of length $n$ over $\\mathbb{Z}_{2^m}$ with respect to a vector $s\\in \\mathbb{Z}_{2^m}^n\\setminus \\mathcal{C}$ satisfying either $s\\cdot s \\equiv 0 (mod 2^m)$ or $s\\cdot s \\equiv 2^{m-1} (mod 2^m).$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-14T20:07:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "construction",
      "higher lengths",
      "jacobi forms",
      "positive integer",
      "construct self-dual codes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.4827S"
    }
  }
}