An Upper Bound for the Circumference of a 3-Connected Binary Matroid

Jim Geelen and Peter Nelson proved that, for a loopless connected binary matroid $M$ with an odd circuit, if a largest odd circuit of $M$ has $k$ elements, then a largest circuit of $M$ has at most $2k-2$ elements. The goal of this note is to show that, when $M$ is $3$-connected, either $M$ has a spanning circuit, or a largest circuit of $M$ has at most $2k-4$ elements. Moreover, the latter holds when $M$ is regular of rank at least four.