  Las Vergnas and Hamidoune studied the number of circuits needed to determine
an oriented matroid. In this paper we investigate this problem and some new
variants, as well as their interpretation in particular classes of matroids. We
present general upper and lower bounds in the setting of general connected
orientable matroids, leading to the study of subgraphs of the base graph and
the intersection graph of circuits.
  We then consider the problem for uniform matroids which is closely related to
the notion of (connected) covering numbers in Design Theory. Finally, we also
devote special attention to regular matroids as well as some graphic and
cographic matroids leading in particular to the topics of (connected) bond and
cycle covers in Graph Theory.
