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

07/13/2018

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by   Rohit Gurjar, et al.

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