{
  "id": "1602.02531",
  "version": "v1",
  "published": "2016-02-08T11:30:19.000Z",
  "updated": "2016-02-08T11:30:19.000Z",
  "title": "Semidefinite bounds for nonbinary codes based on quadruples",
  "authors": [
    "Bart Litjens",
    "Sven Polak",
    "Alexander Schrijver"
  ],
  "categories": [
    "math.CO",
    "math.OC",
    "math.RT"
  ],
  "abstract": "For nonnegative integers $q,n,d$, let $A_q(n,d)$ denote the maximum cardinality of a code of length $n$ over an alphabet $[q]$ with $q$ letters and with minimum distance at least $d$. We consider the following upper bound on $A_q(n,d)$. For any $k$, let $\\CC_k$ be the collection of codes of cardinality at most $k$. Then $A_q(n,d)$ is at most the maximum value of $\\sum_{v\\in[q]^n}x(\\{v\\})$, where $x$ is a function $\\CC_4\\to R_+$ such that $x(\\emptyset)=1$ and $x(C)=0$ if $C$ has minimum distance less than $d$, and such that the $\\CC_2\\times\\CC_2$ matrix $(x(C\\cup C'))_{C,C'\\in\\CC_2}$ is positive semidefinite. By the symmetry of the problem, we can apply representation theory to reduce the problem to a semidefinite programming problem with order bounded by a polynomial in $n$. It yields the new upper bounds $A_4(6,3)\\leq 176$, $A_4(7,4)\\leq 155$, $A_5(7,4)\\leq 489$, and $A_5(7,5)\\leq 87$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-08T11:30:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "94B65",
      "05E10",
      "90C22",
      "20C30"
    ],
    "keywords": [
      "semidefinite bounds",
      "nonbinary codes",
      "quadruples",
      "minimum distance",
      "upper bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160202531L"
    }
  }
}