In this work we show that algebraic lattices con-
structed from error-correcting codes achieve the ergodic capacity
of the fading channel. The main ingredients for our construction
are a generalized version of the Minkowski-Hlawka theorem and
shaping techniques based on the lattice Gaussian distribution.
The structure of the ring of integers in a number field plays
an important role in the proposed construction. In the case
of independent and identically distributed fadings, the lattices
considered exhibit full diversity and an exponential decay of the
probability of error with respect to the blocklength.