Some "Goodness" Properties of LDA Lattices

TL;DR

This paper investigates the structural properties of LDA lattices derived from LDPC codes, demonstrating their effectiveness for various communication channel capacities and their suitability for low-complexity decoding.

Contribution

It proves that LDA lattices are good for packing and quantization, and their duals are good for packing, extending their applicability to multiple communication scenarios.

Findings

LDA lattices are good for packing and MSE quantization.

Their duals are good for packing.

Nested LDA lattices can achieve multiple channel capacities.

Abstract

We study some structural properties of Construction-A lattices obtained from low density parity check (LDPC) codes over prime fields. Such lattices are called low density Construction-A (LDA) lattices, and permit low-complexity belief propagation decoding for transmission over Gaussian channels. It has been shown that LDA lattices achieve the capacity of the power constrained additive white Gaussian noise (AWGN) channel with closest lattice-point decoding, and simulations suggested that they perform well under belief propagation decoding. We continue this line of work, and prove that these lattices are good for packing and mean squared error (MSE) quantization, and that their duals are good for packing. With this, we can conclude that codes constructed using nested LDA lattices can achieve the capacity of the power constrained AWGN channel, the capacity of the dirty paper channel, the rates guaranteed by the compute-and-forward protocol, and the best known rates for bidirectional relaying with perfect secrecy.