Some “goodness” properties of LDA lattices

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 have been shown to achieve the capacity of the AWGN channel under closest lattice-point decoding. Also, simulations suggest that they perform well under belief propagation decoding. In this work, we prove that LDA 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 capacities of the AWGN channel and the dirty paper channel, the rates guaranteed by the compute-and-forward protocol, and the best known rates for bidirectional relaying with perfect secrecy.