Large Circuits in Binary Matroids of Large Cogirth, II

Let F 7 denote the Fano matroid and M be a simple connected binary matroid such that every cocircuit of M has size at least d 3. We show that if M does not have an F 7-minor, M 6 = F 7 , and d = 2 f5; 6; 7; 8g, then M has a circuit of size at least minfr(M) + 1; 2dg. We conjecture that the latter result holds for all d 3.