On the circuit-spectrum of binary matroids

Murty, in 1971, characterized the connected binary matroids with all circuits having the same size. We characterize the connected binary matroids with circuits of two different sizes, where the largest size is odd. As a consequence of this result we obtain both Murty’s result and other results on binary matroids with circuits of only two sizes. We also show that it will be difficult to complete the general case of this problem.