{
  "id": "1606.06930",
  "version": "v1",
  "published": "2016-06-22T12:50:08.000Z",
  "updated": "2016-06-22T12:50:08.000Z",
  "title": "Semidefinite bounds for mixed binary/ternary codes",
  "authors": [
    "Bart Litjens"
  ],
  "comment": "12 pages",
  "categories": [
    "math.CO",
    "math.OC",
    "math.RT"
  ],
  "abstract": "For nonnegative integers $n_2, n_3$ and $d$, let $N(n_2,n_3,d)$ denote the maximum cardinality of a code of length $n_2+n_3$, with $n_2$ binary coordinates and $n_3$ ternary coordinates (in this order) and with minimum distance at least $d$. For a nonnegative integer $k$, let $\\mathcal{C}_k$ denote the collection of codes of cardinality at most $k$. For $D \\in \\mathcal{C}_k$, define $S(D) := \\{C \\in \\mathcal{C}_k \\mid D \\subseteq C, |D| +2|C\\setminus D| \\leq k\\}$. Then $N(n_2,n_3,d)$ is upper bounded by the maximum value of $\\sum_{v \\in [2]^{n_2}[3]^{n_3}}x(\\{v\\})$, where $x$ is a function $\\mathcal{C}_k \\rightarrow \\mathbb{R}$ such that $x(\\emptyset) = 1$ and $x(C) = 0$ if $C$ has minimum distance less than $d$, and such that the $S(D)\\times S(D)$ matrix $(x(C\\cup C'))_{C,C' \\in S(D)}$ is positive semidefinite for each $D \\in \\mathcal{C}_k$. By exploiting symmetry, the semidefinite programming problem for the case $k=3$ is reduced using representation theory. It yields $134$ new upper bounds that are provided in tables",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-22T12:50:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "94B65",
      "05E10",
      "90C22",
      "20C30"
    ],
    "keywords": [
      "mixed binary/ternary codes",
      "semidefinite bounds",
      "minimum distance",
      "nonnegative integer",
      "maximum value"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}