Low-Dimensional Shaping for High-Dimensional Lattice Codes

TL;DR

This paper introduces two low-complexity lattice code constructions that achieve high shaping and coding gains, suitable for point-to-point and compute-and-forward channels, with improved performance and lower complexity.

Contribution

It proposes two novel lattice shaping methods—systematic Voronoi shaping and mixed nested lattice codes—that enhance gains and provide algebraic structure for advanced communication applications.

Findings

Shaping gain of up to 0.86 dB achieved

Lower complexity than existing methods

Effective for point-to-point and compute-and-forward channels

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 (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.