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PreprintCC BY 4.0arXiv

A counterexample to the circuit-cocircuit intersection conjecture

  • Houshan Fu
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Abstract

Oxley conjectured that a matroid with a circuit-cocircuit intersection of size k≥ 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.

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Cite

BibTeXRIS
BibTeX
@misc{fu2026counterexample,
  title = {{A counterexample to the circuit-cocircuit intersection conjecture}},
  author = {Houshan Fu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.06033},
  eprint = {2610.06033},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.06033},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A counterexample to the circuit-cocircuit intersection conjecture
AU  - Houshan Fu
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.06033
UR  - https://arxiv.org/abs/2610.06033
AB  - Oxley conjectured that a matroid with a circuit-cocircuit intersection of size k≥ 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.
ER  -