{
  "id": "math/0312467",
  "version": "v1",
  "published": "2003-12-26T17:44:34.000Z",
  "updated": "2003-12-26T17:44:34.000Z",
  "title": "Nonintersecting Subspaces Based on Finite Alphabets",
  "authors": [
    "Frederique E. Oggier",
    "N. J. A. Sloane",
    "A. R. Calderbank",
    "Suhas N. Diggavi"
  ],
  "comment": "14 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Two subspaces of a vector space are here called ``nonintersecting'' if they meet only in the zero vector. The following problem arises in the design of noncoherent multiple-antenna communications systems. How many pairwise nonintersecting M_t-dimensional subspaces of an m-dimensional vector space V over a field F can be found, if the generator matrices for the subspaces may contain only symbols from a given finite alphabet A subseteq F? The most important case is when F is the field of complex numbers C; then M_t is the number of antennas. If A = F = GF(q) it is shown that the number of nonintersecting subspaces is at most (q^m-1)/(q^{M_t}-1), and that this bound can be attained if and only if m is divisible by M_t. Furthermore these subspaces remain nonintersecting when ``lifted'' to the complex field. Thus the finite field case is essentially completely solved. In the case when F = C only the case M_t=2 is considered. It is shown that if A is a PSK-configuration, consisting of the 2^r complex roots of unity, the number of nonintersecting planes is at least 2^{r(m-2)} and at most 2^{r(m-1)-1} (the lower bound may in fact be the best that can be achieved).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-12-26T17:44:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11T22",
      "11T71",
      "51E23",
      "94A99"
    ],
    "keywords": [
      "finite alphabet",
      "nonintersecting subspaces",
      "noncoherent multiple-antenna communications systems",
      "m-dimensional vector space",
      "finite field case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math.....12467O"
    }
  }
}