On the Number of Circuits in Regular Matroids (with Connections to Lattices and Codes)

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.