{
  "abstract": "A length-$n$ binary $k$-deletion code is a set of binary strings such that if we delete any $k$ bits of a string, leaving a length-$(n-k)$ binary string, we can uniquely recover the codeword. In this paper, we consider $t$-list decodable deletion codes, where after $k$ bits of a codeword are deleted, we can identify a list of size at most $t$ such that the original codeword lies in the list. We prove a lower bound of $Ω_k(2^n t\\log^{1/t}n/n^{k+k/t})$ on the optimal size of a $t$-list decodable $k$-deletion code, giving a $\\sqrt{\\log n}$ improvement over the previously best known bounds for $2$-list decodable $2$-deletion codes [GH21] and providing the first nontrivial lower bound when $t>2$ or $k>2$. Our bound holds for all $t\\leq n^k$, showing that $t=Ω(\\log n)-$list decodable deletion codes have optimal size $Θ_k(2^n t/n^k)$, asymptotically matching the known upper bound. We also prove upper bounds on the number of common subsequences and common supersequences of a given length for any two binary strings.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26650",
  "author_total": 1,
  "authors": [
    {
      "name": "Andrew D. Lin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fk6y2stdx255x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26650",
  "fulltext_url": null,
  "id": "fk6y2stdx255x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26650",
  "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",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7214127811",
  "pdf_url": "https://arxiv.org/pdf/2609.26650",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fk6y2stdx255x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26650",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26650"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Lower Bounds for all List-Decodable Deletion Codes",
  "title_text": "Lower Bounds for all List-Decodable Deletion Codes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/fk6y2stdx255x",
  "venue": "arXiv",
  "versions": []
}