The circuit-cocircuit intersection conjecture asserts that, if a matroid circuit and a matroid cocircuit of a matroid have an intersection
of size 
, then some matroid
circuit and matroid cocircuit have an intersection of size 
(Fu 2026).
Fu (2026) constructed a counterexample using the binary matroid represented by a generator matrix of the extended
binary Golay code. This matroid
has matroid rank 12 and 24 elements and is equal
to its dual matroid. The possible sizes of its matroid circuit-matroid
cocircuit intersections are exactly 
. In particular, an intersection
of size 12 exists, but none has size 10.
