{
  "id": "1809.02960",
  "version": "v1",
  "published": "2018-09-09T12:19:38.000Z",
  "updated": "2018-09-09T12:19:38.000Z",
  "title": "Laplacian Simplices II: A Coding Theoretic Approach",
  "authors": [
    "Marie Meyer",
    "Tefjol Pllaha"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "This paper further investigates \\emph{Laplacian simplices}. A construction by Braun and the first author associates to a simple connected graph $G$ a simplex $\\cP_G$ whose vertices are the rows of the Laplacian matrix of $G$. In this paper we associate to a reflexive $\\cP_G$ a duality-preserving linear code $\\cC(\\cP_G)$. This new perspective allows us to build upon previous results relating graphical properties of $G$ to properties of the polytope $\\cP_G$. In particular, we make progress towards a graphical characterization of reflexive $\\cP_G$ using techniques from Ehrhart theory. We provide a systematic investigation of $\\cC(\\cP_G)$ for cycles, complete graphs, and graphs with a prime number of vertices. We construct an asymptotically good family of MDS codes. In addition, we show that any rational rate is achievable by such construction.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-09T12:19:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "52B20",
      "94B05"
    ],
    "keywords": [
      "coding theoretic approach",
      "laplacian simplices",
      "first author associates",
      "construction",
      "rational rate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}