Binary matroids with no 4-spike minors

For a simple binary matroid M having no n -spike minor, we examine the problem of bounding | E ( M ) | as a function of its rank r ( M ) and circumference c ( M ) . In particular, we show that | E ( M ) | ≤ min { r ( M ) ( r ( M ) + 3 ) 2 , c ( M ) r ( M ) } for any simple, binary matroid M having no 4-spike minor. As a consequence, the same bound applies to simple, binary matroids having no A G ( 3 , 2 ) -minor.