{
  "abstract": "We study the problem of determining upper bounds on the maximal binary rank, $b_d$, of a $0,1$ matrix with real rank $d$, for small constant $d$. We prove a general upper bound of $b_d \\leq \\left\\lfloor \\sqrt{d2^d + 2d}\\right\\rfloor -1$. We also develop a general dimension reduction technique of bounding $b_d$ using known upper bounds on $b_{d-1}$ and $b_{d-2}$, as well as prove interesting general structural results on extremal matrices with binary rank $b_d$. This allows us to get improved non-trivial upper bounds on $b_d$ for constant values of $d$, which imply tight bounds on $b_d$ for $3 \\leq d \\leq 5$. Our results also imply bounds on the equivalent problem of finding the minimum number of bicliques needed to partition the edges of a bipartite graph whose reduced adjacency matrix has real rank at most $d$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30203",
  "author_total": 1,
  "authors": [
    {
      "name": "Michal Parnas"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/04k0nmnzrbc4b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30203",
  "fulltext_url": null,
  "id": "04k0nmnzrbc4b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30203",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7214361386",
  "pdf_url": "https://arxiv.org/pdf/2609.30203",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/04k0nmnzrbc4b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2609.30203",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30203"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On the Binary Rank of Matrices with Constant Real Rank",
  "title_text": "On the Binary Rank of Matrices with Constant Real Rank",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/04k0nmnzrbc4b",
  "venue": "arXiv",
  "versions": []
}