  We propose two low-complexity lattice code constructions that have
competitive coding and shaping gains. The first construction, named systematic
Voronoi shaping, maps short blocks of integers to the dithered Voronoi
integers, which are dithered integers that are uniformly distributed over the
Voronoi region of a low-dimensional shaping lattice. Then, these dithered
Voronoi integers are encoded using a high-dimensional lattice retaining the
same shaping and coding gains of low and high-dimensional lattices. A drawback
to this construction is that there is no isomorphism between the underlying
message and the lattice code, preventing its use in applications such as
compute-and- forward. Therefore we propose a second construction, called mixed
nested lattice codes, in which a high-dimensional coding lattice is nested
inside a concatenation of low-dimensional shaping lattices. This construction
not only retains the same shaping/coding gains as first construction but also
provides the desired algebraic structure. We numerically study these methods,
for point-to-point channels as well as compute-and-forward using low-density
lattice codes (LDLCs) as coding lattices and E8 and Barnes-Wall as shaping
lattices. Numerical results indicate a shaping gain of up to 0.86 dB, compared
to the state-of-the-art of 0.4 dB; furthermore, the proposed method has lower
complexity than state-of-the-art approaches.
