{
  "abstract": "Data stored in synthetic DNA is retrieved by shotgun sequencing, which returns short subsequences rather than the stored word itself. A natural abstraction of this readout is the $k$-deck of a word: the vector recording how often each word of length $k$ occurs as a subsequence. Two stored words are distinguishable from their readouts exactly when their $k$-decks differ, so the number $D_{q,k}(n)$ of distinct $k$-decks of words of length $n$ over an alphabet of size $q$ measures what a length-$k$ readout retains. We analyse the degrees of freedom remaining in a $k$-deck once all shorter decks are fixed. Within each class of words having prescribed letter multiplicities, the length-$k$ entries are confined to an affine subspace whose dimension is exactly the number of Lyndon words with the same multiplicities, which we give in closed form as a Möbius sum. Writing $L_q(j)$ for the number of Lyndon words of length $j$ over an alphabet of size $q$, we deduce the improved upper bound \\[ D_{q,k}(n)=O\\!\\left(n^{E_q(k)}\\right),\\qquad E_q(k)=\\sum_{j=1}^{k}j\\,L_q(j)-1 . \\] In the case of a binary alphabet this bound satisfies $D_{2,k}(n)=O\\!\\left(n^{4\\cdot 2^{k-1}}\\right)$. We then prove matching lower bounds in the first two nontrivial cases: $D_{q,2}(n)=Θ\\!\\left(n^{q^2-1}\\right)$ for every alphabet size $q$, and $D_{2,3}(n)=Θ(n^{9})$ for the binary alphabet. The latter confirms, for $q=2$ and $k=3$, our conjecture that the upper bound has the correct degree for every $q$ and $k$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23106",
  "author_total": 2,
  "authors": [
    {
      "name": "Arman Nilforoushan"
    },
    {
      "name": "Farzad Parvaresh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6n19r7hz7fk28.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23106",
  "fulltext_url": null,
  "id": "6n19r7hz7fk28",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23106",
  "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.IT"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Algorithms and Data Compression"
    ]
  ],
  "openalex": "W7214053605",
  "pdf_url": "https://arxiv.org/pdf/2609.23106",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6n19r7hz7fk28.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.23106",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23106"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Improved upper bound on the number of distinct k-decks for any k and alphabet size by counting the independent parameters",
  "title_text": "Improved upper bound on the number of distinct k-decks for any k and alphabet size by counting the independent parameters",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/6n19r7hz7fk28",
  "venue": "arXiv",
  "versions": []
}