  This paper addresses point-to-point communication over block-fading channels
with independent fading blocks. When both channel state information at the
transmitter (CSIT) and receiver (CSIR) are available, most achievable schemes
use separable coding, i.e., coding independently and in parallel over different
fading states. Unfortunately, separable coding has drawbacks including large
memory requirements at both communication ends. In this paper a lattice coding
and decoding scheme is proposed that achieves the ergodic capacity without
separable coding, with lattice codebooks and decoding decision regions that are
universal across channel realizations. We first demonstrate this result for
fading distributions with discrete, finite support whose sequences are robustly
typical. Results are then extended to continuous fading distributions, as well
as multiple-input multiple-output (MIMO) systems. In addition, a variant of the
proposed scheme is presented for the MIMO ergodic fading channel with CSIR
only, where we prove the existence of a universal codebook that achieves rates
within a constant gap to capacity for finite-support fading distributions. The
gap is small compared with other schemes in the literature. Extension to
continuous-valued fading is also provided.
