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 AG(3,2)-minor.