{
  "id": "0706.3313",
  "version": "v1",
  "published": "2007-06-22T11:49:07.000Z",
  "updated": "2007-06-22T11:49:07.000Z",
  "title": "Parity properties of Costas arrays defined via finite fields",
  "authors": [
    "Konstantinos Drakakis",
    "Rod Gow",
    "Scott rickard"
  ],
  "comment": "To appear in Advances in Mathematics of Communications",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "A Costas array of order $n$ is an arrangement of dots and blanks into $n$ rows and $n$ columns, with exactly one dot in each row and each column, the arrangement satisfying certain specified conditions. A dot occurring in such an array is even/even if it occurs in the $i$-th row and $j$-th column, where $i$ and $j$ are both even integers, and there are similar definitions of odd/odd, even/odd and odd/even dots. Two types of Costas arrays, known as Golomb-Costas and Welch-Costas arrays, can be defined using finite fields. When $q$ is a power of an odd prime, we enumerate the number of even/even odd/odd, even/odd and odd/even dots in a Golomb-Costas array. We show that three of these numbers are equal and they differ by $\\pm 1$ from the fourth. For a Welch-Costas array of order $p-1$, where $p$ is an odd prime, the four numbers above are all equal to $(p-1)/4$ when $p\\equiv 1\\pmod{4}$, but when $p\\equiv 3\\pmod{4}$, we show that the four numbers are defined in terms of the class number of the imaginary quadratic field $\\mathbb{Q}(\\sqrt{-p})$, and thus behave in a much less predictable manner.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-06-22T11:49:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "costas arrays",
      "finite fields",
      "parity properties",
      "welch-costas array",
      "odd/even dots"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0706.3313D"
    }
  }
}