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A counterexample to the circuit-cocircuit intersection conjecture — arXiv:2610.06033

Source: arXiv (en)
Original: https://arxiv.org/abs/2610.06033
License (as archived): CC0-1.0
Archived: 2026-10-06T16:52:17.650Z
SHA-256 of the archived text: 54d261204d963e25f48dc42fc38034f524079b114875a964a84b2c947d619392
Stable URL of this page: https://metraipolis.com/documento/dc_febdb83dbc52

How to cite

A counterexample to the circuit-cocircuit intersection conjecture — arXiv:2610.06033 — arXiv (en), https://arxiv.org/abs/2610.06033 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_febdb83dbc52 (Metraipolis archive dc_febdb83dbc52, sha256 54d261204d963e25f48dc42fc38034f524079b114875a964a84b2c947d619392)

BibTeX

@misc{dcfebdb83dbc52,
  title = {A counterexample to the circuit-cocircuit intersection conjecture — arXiv:2610.06033},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.06033},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_febdb83dbc52; sha256 54d261204d963e25f48dc42fc38034f524079b114875a964a84b2c947d619392},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.06033. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Oxley conjectured that a matroid with a circuit-cocircuit intersection of size $k\geq 4$ has one of size $k-2$. Using the standard polynomial representation of the classical extended binary Golay code, we construct an identically self-dual binary matroid of rank $12$ on $24$ elements. Its circuit-cocircuit intersection sizes are exactly $0,2,4,6,8$, and $12$, disproving the conjecture for $k=12$. The proof is self-contained and does not rely on the full weight distribution of the code or on an enumeration of all circuits.

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/2610.06033 — 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 2026-10-06T15:19:20.010Z dalla pagina https://arxiv.org/list/math.CO/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).