  In this article we study the Ahlfors regular conformal gauge of a compact
metric space $(X,d)$, and its conformal dimension $\mathrm{dim}_{AR}(X,d)$.
Using a sequence of finite coverings of $(X,d)$, we construct distances in its
Ahlfors regular conformal gauge of controlled Hausdorff dimension. We obtain in
this way a combinatorial description, up to bi-Lipschitz homeomorphisms, of all
the metrics in the gauge. We show how to compute $\mathrm{dim}_{AR}(X,d)$ using
the critical exponent $Q_N$ associated to the combinatorial modulus.
