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The minimum number of minimal codewords in an \([n, k]\)-code and in graphic codes. (English) Zbl 1311.05027
Summary: We survey some lower bounds on the function in the title based on matroid theory and address the following problem by G. Dosa et al. [PU.M.A., Pure Math. Appl. 15, No. 4, 383–392 (2004; Zbl 1112.05021)]: 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 \(p \geq 38\), every 3-connected graph with \(p\) vertices has at least as many cycles as the wheel with \(p\) vertices.

05B35 Combinatorial aspects of matroids and geometric lattices
94B25 Combinatorial codes
Magma; nauty; Traces
Full Text: DOI
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