{
  "id": "2408.16721",
  "version": "v1",
  "published": "2024-08-29T17:14:12.000Z",
  "updated": "2024-08-29T17:14:12.000Z",
  "title": "Modular Golomb rulers and almost difference sets",
  "authors": [
    "Daniel M. Gordon"
  ],
  "comment": "6 pages, 1 figure",
  "categories": [
    "math.CO"
  ],
  "abstract": "A $(v,k,\\lambda)$-difference set in a group $G$ of order $v$ is a subset $\\{d_1, d_2, \\ldots,d_k\\}$ of $G$ such that $D=\\sum d_i$ in the group ring ${\\mathbb Z}[G]$ satisfies $$D D^{-1} = n + \\lambda G,$$ where $n=k-\\lambda$. In other words, the nonzero elements of $G$ all occur exactly $\\lambda$ times as differences of elements in $D$. A $(v,k,\\lambda,t)$-almost difference set has $t$ nonzero elements of $G$ occurring $\\lambda$ times, and the other $v-1-t$ occurring $\\lambda+1$ times. When $\\lambda=0$, this is equivalent to a modular Golomb ruler. In this paper we investigate existence questions on these objects, and extend previous results constructing almost difference sets by adding or removing an element from a difference set.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-29T17:14:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B10"
    ],
    "keywords": [
      "difference set",
      "modular golomb ruler",
      "nonzero elements",
      "existence questions",
      "equivalent"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}