  We study a counting version of Cycle Double Cover Conjecture. We discuss why
it is more interesting to count circuits (i.e., graphs isomorphic to $C_k$ for
some $k$) instead of cycles (graphs with all degrees even). We give an
almost-exponential lower-bound for graphs with a surface embedding of
representativity at least 4. We also prove an exponential lower-bound for
planar graphs. We conjecture that any bridgeless cubic graph has at least
$2^{n/2-1}$ circuit double covers and we show an infinite class of graphs for
which this bound is tight.
