LDA Lattices Without Dithering Achieve Capacity on the Gaussian Channel

TL;DR

This paper proves that LDA lattices constructed without dithering can achieve the Gaussian channel capacity using simple encoding and decoding, with capacity-achieving properties linked to their sparse structure.

Contribution

It demonstrates that LDA lattices without dithering achieve capacity on the AWGN channel, simplifying previous methods and analyzing the behavior of the underlying prime number.

Findings

LDA lattices achieve AWGN channel capacity.

Capacity is achievable with constant-weight parity-check matrices.

Expansion properties of bipartite graphs help bound minimum Euclidean distance.

Abstract

This paper deals with Low-Density Construction-A (LDA) lattices, which are obtained via Construction A from non-binary Low-Density Parity-Check codes. More precisely, a proof is provided that Voronoi constellations of LDA lattices achieve the capacity of the AWGN channel under lattice encoding and decoding. This is obtained after showing the same result for more general Construction-A lattice constellations. The theoretical analysis is carried out in a way that allows to describe how the prime number underlying Construction A behaves as a function of the lattice dimension. Moreover, no dithering is required in the transmission scheme, simplifying some previous solutions of the problem. Remarkably, capacity is achievable with LDA lattice codes whose parity-check matrices have constant row and column Hamming weights. Some expansion properties of random bipartite graphs constitute an extremely important tool for dealing with sparse matrices and allow to find a lower bound of the minimum Euclidean distance of LDA lattices in our ensemble.