The minimum number of minimal codewords in an [n, k]-code and in graphic codes

We survey some lower bounds on the function in the title based on matroid theory and address the following problem by Dosa, Szalkai, Laflamme [9]: Determine the smallest number of circuits in a loopless matroid with no parallel elements and with a given size and rank. In the graphic 3-connected case we provide a lower bound which is a product of a linear function of the number of vertices and an exponential function of the average degree. We also prove that, for p ≥ 38, every 3-connected graph with p vertices has at least as many cycles as the wheel with p vertices.