{
  "id": "1907.04391",
  "version": "v1",
  "published": "2019-07-09T20:17:38.000Z",
  "updated": "2019-07-09T20:17:38.000Z",
  "title": "Some constructions of quantum MDS codes",
  "authors": [
    "Simeon Ball"
  ],
  "categories": [
    "quant-ph",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We construct quantum MDS codes for quantum systems of dimension $q$ of length $q^2+1$ and minimum distance $d$ for all $d \\leqslant q+1$, $d \\neq q$. These codes are shown to exist by proving that there are classical generalised Reed-Solomon codes which are contained in their Hermitian-dual. These constructions include many constructions which were previously known but in some cases these codes appear to be new. We go on to prove that if $d\\geqslant q+2$ then there in no generalised Reed-Solomon code which is contained in its Hermitian dual. We also construct a $ [\\![ 18,0,10 ]\\!] _5$ quantum MDS code, a $ [\\![ 18,0,10 ]\\!] _7$ quantum MDS code and a $ [\\![ 14,0,8 ]\\!] _5$ quantum MDS code, which are the first quantum MDS codes discovered for which $d \\geqslant q+3$, apart from the $ [\\![ 10,0,6 ]\\!] _3$ quantum MDS code derived from Glynn's code.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-09T20:17:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "constructions",
      "first quantum mds codes",
      "construct quantum mds codes",
      "quantum systems",
      "hermitian dual"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}