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

TL;DR

This paper establishes bounds on the number of circuits, shortest vectors, and minimum weight codewords in regular matroids, lattices, and codes, generalizing known results and connecting combinatorial and algebraic structures.

Contribution

It extends bounds on $ ext{α}$-minimum circuits to regular matroids, lattices, and codes, unifying combinatorial and algebraic perspectives.

Findings

Bound on the number of α-minimum circuits in regular matroids as m^{O(α^2)}

Similar bounds on α-shortest vectors in totally unimodular lattices

Bounds on α-minimum weight codewords in regular codes

Abstract

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.