  We show that for any regular matroid on $m$ elements and any $\alpha \geq 1$,
the number of $\alpha$-minimum circuits, or circuits whose size is at most an
$\alpha$-multiple of the minimum size of a circuit in the matroid is bounded by
$m^{O(\alpha^2)}$. This generalizes a result of Karger for the number of
$\alpha$-minimum cuts in a graph. As a consequence, we obtain similar bounds on
the number of $\alpha$-shortest vectors in "totally unimodular" lattices and on
the number of $\alpha$-minimum weight codewords in "regular" codes.
