We first determine the asymptotes of the epsilon-covering numbers of Holder-Zygmund type spaces on data-defined manifolds. Secondly, a fully discrete and finite algorithmic scheme is developed providing explicit epsilon-coverings whose cardinality is asymptotically near the epsilon-covering number. Given an arbitrary Holder-Zygmund type function, the nearby center of a ball in the epsilon-covering can also be computed in a discrete finite fashion.