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We propose two low-complexity lattice code constructions that havecompetitive coding and shaping gains. The first construction, named systematicVoronoi shaping, maps short blocks of integers to the dithered Voronoiintegers, which are dithered integers that are uniformly distributed over theVoronoi region of a low-dimensional shaping lattice. Then, these ditheredVoronoi integers are encoded using a high-dimensional lattice retaining thesame shaping and coding gains of low and high-dimensional lattices. A drawbackto this construction is that there is no isomorphism between the underlyingmessage and the lattice code, preventing its use in applications such ascompute-and- forward. Therefore we propose a second construction, called mixednested lattice codes, in which a high-dimensional coding lattice is nestedinside a concatenation of low-dimensional shaping lattices. This constructionnot only retains the same shaping/coding gains as first construction but alsoprovides the desired algebraic structure. We numerically study these methods,for point-to-point channels as well as compute-and-forward using low-densitylattice codes (LDLCs) as coding lattices and E8 and Barnes-Wall as shapinglattices. Numerical results indicate a shaping gain of up to 0.86 dB, comparedto the state-of-the-art of 0.4 dB; furthermore, the proposed method has lowercomplexity than state-of-the-art approaches.

Low-Dimensional Shaping for High-Dimensional Lattice Codes