  We present constructions of Space-Time (ST) codes based on lattice coset
coding. First, we focus on ST code constructions for the short block-length
case, i.e., when the block-length is equal to or slightly larger than the
number of transmit antennas. We present constructions based on dense lattice
packings and nested lattice (Voronoi) shaping. Our codes achieve the optimal
diversity-multiplexing tradeoff of quasi-static MIMO fading channels for any
fading statistics, and perform very well also at practical, moderate values of
signal to noise ratios (SNR). Then, we extend the construction to the case of
large block lengths, by using trellis coset coding. We provide constructions of
trellis coded modulation (TCM) schemes that are endowed with good packing and
shaping properties. Both short-block and trellis constructions allow for a
reduced complexity decoding algorithm based on minimum mean squared error
generalized decision feedback equalizer (MMSE-GDFE) lattice decoding and a
combination of this with a Viterbi TCM decoder for the TCM case. Beyond the
interesting algebraic structure, we exhibit codes whose performance is among
the state-of-the art considering codes with similar encoding/decoding
complexity.
