Dugundji's canonical covers, asymptotic and covering dimension

Given a nowheredense closed subset X of a metrizable compact space X˜, we characterize the dimension of X in terms of the multiplicity of the canonical covers of the complementary of X, especially in some particular cases, like when X˜ is the Hilbert cube or the finite dimensional cube and X, a Z-set of X˜. In this process, we solve some related questions in the literature.