{
  "abstract": "The maximum size of a t-intersecting code in the set of words of length n over an alphabet of size m, a problem posed by Frankl and Füredi, was determined independently by Ahlswede and Khachatrian and by Frankl and Tokushige. Extending this result to the forbidden-intersection setting, Keevash, Lifshitz, Long and Minzer proved that, for m greater than two and n sufficiently large in terms of t, (t minus one)-avoiding codes have the same maximum size and the same extremal ball constructions as t-intersecting codes. They also raised the binary case. Here, t-intersecting requires every two distinct words to agree in at least t coordinates, while (t minus one)-avoiding forbids agreement in exactly t minus one coordinates. We solve the binary case for all t at least three and n at least the square of the sum of t and one. If the forbidden Hamming distance n minus t plus one is odd, the maximum size of a (t minus one)-avoiding code strictly exceeds the maximum t-intersecting size. If it is even, the two maxima coincide, but their extremal families differ: some maximum (t minus one)-avoiding codes are not t-intersecting. We also determine all maximum avoiding codes in the even case. For odd forbidden distances, weight parity and a fixed-point-free pairing determine the maximum, while the diameter theorem gives the comparison with t-intersecting codes. For even forbidden distances, parity splitting and coordinate deletion reduce the problem to avoiding two consecutive distances. Connected components, disjoint neighbourhoods, and Harper's vertex-isoperimetric theorem yield the upper bound, which is attained by a t-intersecting construction. The diameter theorem then determines the extremal structure, and elementary binomial estimates give the stated parameter range.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.04523",
  "author_total": 2,
  "authors": [
    {
      "name": "Jianwei Jiang"
    },
    {
      "name": "Chunhua Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4ggfgcgmmxtnx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.04523",
  "fulltext_url": null,
  "id": "4ggfgcgmmxtnx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.04523",
  "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",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.04523",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4ggfgcgmmxtnx.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.04523",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.04523"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Binary Codes with a Forbidden Distance",
  "title_text": "Binary Codes with a Forbidden Distance",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/4ggfgcgmmxtnx",
  "venue": "arXiv",
  "versions": []
}