Abstract
We show that for any 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 mO(α2). 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.
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