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Binary Codes with a Forbidden Distance — arXiv:2611.04523

Source: arXiv (en)
Original: https://arxiv.org/abs/2611.04523
License (as archived): CC0-1.0
Archived: 2026-11-04T12:48:47.216Z
SHA-256 of the archived text: 88b3ed31cccd6edda0bbc997ac50d9b0c432c66ed6cbe4fe843c923c35a444c8
Stable URL of this page: https://metraipolis.com/documento/dc_8917011fee90

How to cite

Binary Codes with a Forbidden Distance — arXiv:2611.04523 — arXiv (en), https://arxiv.org/abs/2611.04523 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_8917011fee90 (Metraipolis archive dc_8917011fee90, sha256 88b3ed31cccd6edda0bbc997ac50d9b0c432c66ed6cbe4fe843c923c35a444c8)

BibTeX

@misc{dc8917011fee90,
  title = {Binary Codes with a Forbidden Distance — arXiv:2611.04523},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2611.04523},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_8917011fee90; sha256 88b3ed31cccd6edda0bbc997ac50d9b0c432c66ed6cbe4fe843c923c35a444c8},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2611.04523. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.CO, nuovi lavori del giorno).
Categorie arXiv: Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2611.04523 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-11-04T12:32:32.030Z dalla pagina https://arxiv.org/list/math.CO/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).