Some “goodness” properties of LDA lattices

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.This work was presented in part at the 2015 IEEE Information Theory Workshop at Jerusalem, Israel. Shashank Vatedka and Navin Kashyap are with the

Practical Encoder and Decoder for Power Constrained QC LDPC-Lattice Codes

“…Proposed typical value for K is 3 and for M is 2 or 6. The decoding complexity of LDA lattices is O(n · d · t · p · log 2 (p)), where p is the characteristic of the finite field that the underlying code is coming from [14], [30]. The least proposed value of p is 11.…”

“…Integer low-density lattices based on construction A which are known as LDA lattices [14] and polar lattices [28], are another families of lattices with practical decoding methods. By applying non-binary LDPC codes and Construction A, LDA lattice of dimension 10000 have obtained an error performance within 0.7dB of Poltyrev's limit [14], [30]. These lattices are shown to be capacity-achieving even without using dithering technique [32].…”

Polar lattices are good for lossy compression

“…Let y ′1:N denotes the realization of N i.i.d copies of source Y ′ . Although Y ′ is a continuous source with density given by (10) and y ′ is drawn from R, to keep the notations consistent (the definition of the Bhattacharyya parameter [26] is given by a summation), from now on we will express the distortion measurement as well as the variantional distance in the form of summation instead of integral.…”

“…Despite enjoying a good practical performance, a theoretical proof of the rate-distortion bound achieving TCQ with low complexity is still missing. More recently, a scheme based on low density Construction-A (LDA) lattices [10] was proved to be quantization-good (defined in Sect. III) using the minimum-distance lattice decoder.…”

“…[25]): If ǫ = ǫ Λ (σ ∆ ) < 1 2 , the variational distance between the density f Y ′ of source Y ′ defined in(10) and the Gaussian density f Y satisfies…”