The maximum number of minimal codewords in long codes

Upper bounds on the maximum number of minimal codewords in a binary code follow from the theory of matroids. Random coding provide lower bounds. In this paper we compare these bounds with analogous bounds for the cycle code of graphs. This problem (in the graphic case) was considered in 1981 by Entringer and Slater who asked if a connected graph with p vertices and q edges can have only slightly more that 2q−p cycles. The bounds in this note answer this in the affirmative for all graphs except possibly some that have fewer than 2p + 3 log2(3p) edges. We also conclude that an Eulerian (even) graph has at most 2q−p cycles unless the graph is a subdivision of a 4-regular graph that is the edge-disjoint union of two Hamiltonian cycles, in which case it may have as many as 2q−p + p cycles.