AbstractMotivated by some problems which had been left open in a previous paper [M. Tarsi, J. Combin. Theory Ser. B 39 (1985), 346–352], we present the following results: 1.1. Every bridgeless binary matroid with no F7∗ minor (in particular every regular matroid) had a cycle in which every element is covered exactly 4 times.2.2. The cycle double cover conjecture for graphs is equivalent to a simular conjecture for binary matroids with no F7∗ minor.3.3. We give the lowest upper bounds for the length of the shortest cycle cover of cographic matroids which admit a k-nowhere zero flow.4.4. We generalize the concept of nowhere zero flows to binary, non-regular matroids and relate it to the length of a shortest cycle cover.