Low-Dimensional Shaping for High-Dimensional Lattice Codes

Abstract

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 as coding lattices and $E_{8}$ and Barnes–Wall as shaping lattices. Numerical results indicate a shaping gain of up to 0.86 dB, compared with the state-of-the-art of 0.4 dB; furthermore, the proposed method has lower complexity than the state-of-the-art approaches.