An upper bound for the circumference of a 3-connected binary matroid (Q2692185)

Summary: \textit{J. Geelen} and \textit{P. Nelson} [Combinatorica 37, No. 1, 41--47 (2017; Zbl 1399.05026)] 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.