{
  "abstract": "We reduce the leading redundancy coefficient for binary codes correcting any fixed number $t\\ge2$ of deletions from $2t$ to $2t-1$. This is the first improvement to that coefficient since Levenshtein's 1965 bound. More generally, over every fixed alphabet of size $q\\ge2$, codes of length $n$ correcting any combination of at most $t$ insertions, deletions and substitutions exist with redundancy $(2t-1)\\log_q n+O_t(\\log_q\\log_q n)$. For two deletions, the upper-bound coefficient drops from $4$ to $3$, halving the gap to the lower-bound coefficient $2$. The leading coefficient $2t-1$ also holds for $t$ bounded substring edits, where both the removed and inserted strings have bounded length. The proof combines random binary linear hashing of substring counts with a two-colouring argument. A linear dependence among differences of substring-count vectors forces their local changes to overlap, and this overlap lowers the redundancy coefficient from $2t$ to $2t-1$. The result provides an edit-distance analogue of Varshamov's 1957 bound in the Hamming metric, both in the leading redundancy coefficient and in the role of linear independence in the proof.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19493",
  "author_total": 1,
  "authors": [
    {
      "name": "Eyal En Gad"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a6rh7b2dkxjk1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19493",
  "fulltext_url": null,
  "id": "a6rh7b2dkxjk1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19493",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.CO"
    ],
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      "arxiv",
      "math.IT"
    ],
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      "openalex",
      "1702|Coding theory and cryptography"
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  "openalex": "W7213581263",
  "pdf_url": "https://arxiv.org/pdf/2609.19493",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a6rh7b2dkxjk1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2609.19493",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19493"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Polynomially larger deletion codes by linear hashing of substring counts",
  "title_text": "Polynomially larger deletion codes by linear hashing of substring counts",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/a6rh7b2dkxjk1",
  "venue": "arXiv",
  "versions": []
}