A rank bound for bases and circuits in binary matroids

Let \(b(M)\), \(d(M)\), and \(r(M)\) denote the number of bases, the number of circuits, and the rank of a matroid \(M\), respectively. We prove that every nonempty simple binary matroid with no coloops satisfies \[ 2b(M)\ge(r(M)+1)d(M), \] with equality if and only if \(M\) is isomorphic to the F