  Given a simple Eulerian binary matroid $M$, what is the minimum number of
disjoint circuits necessary to decompose $M$? We prove that $|M| /
(\operatorname{rank}(M) + 1)$ many circuits suffice if $M = \mathbb F_2^n
\setminus \{0\}$ is the complete binary matroid, for certain values of $n$, and
that $\mathcal{O}(2^{\operatorname{rank}(M)} / (\operatorname{rank}(M) + 1))$
many circuits suffice for general $M$. We also determine the asymptotic
behaviour of the minimum number of circuits in an odd-cover of $M$.
