New results on low-density integer lattices

Abstract

Abstract-A new family of integer lattices built from Construction A and non-binary low-density parity-check (LDPC) codes has been proposed by the authors in 2012. Lattices in this family are referred to as LDA lattices. Previous experimental results revealed excellent performance which clearly single out LDA lattices among the strongest candidates for potential applications in digital communications and networks, such as network coding and information theoretic security at the physical layer level. In this paper, we show that replacing random codes by LDPC codes in Construction A does not induce any structural loss. More precisely, our main theorem states that LDA lattices can achieve Poltyrev capacity limit on an additive white Gaussian noise channel. We present here the detailed proof and its consequences on the lattice dimension, the finite field size, and the parameters of the LDPC ensemble. The latter has a row weight that increases logarithmically in the code length. In a more recent work, it is proved that the Poltyrev limit is attained by a different LDA ensemble having a small constant row weight.

Achieving the AWGN channel capacity with lattice Gaussian coding

“…More precisely, the code book has a discrete Gaussian distribution over an AWGN-good lattice. So the remaining problem is the construction of AWGN-good lattices, which is nonetheless beyond the scope of this paper (see e.g., [5][6][7][8][9] for recent progresses which have approached the Poltyrev capacity). Intuitively, since only shaping is lacking in Poltyrev's technique, the probabilistic shaping inherent with lattice Gaussian distribution will enable it to achieve the AWGN channel capacity.…”

Some “goodness” properties of LDA lattices

“…Large values of p translate to higher BP decoding complexity, and it would be useful to study the structural properties of LDA lattices over fields of smaller sizes. Empirical results by [7], [9] suggest that it may be possible to get good error performance over the AWGN channel (without power constraints) even with moderate field sizes. This suggests that it may be possible to tighten the arguments presented in this paper and obtain better bounds on the parameters of the LDA lattices needed to guarantee the desired properties.…”

“…Simulation results in [7], [27] showed that these lattices perform well with BP decoding. While there is no formal proof that these lattices are good under BP decoding, it was proved in [9] that LDA lattices are good for AWGN channel coding, and subsequently shown in [7], [11] that nested LDA lattices achieve the capacity of the power constrained AWGN channel with closest lattice-point decoding. In this paper, we show that LDA lattices have several other goodness properties.…”

Construction of full-diversity 1-level LDPC lattices for block-fading channels

“…Applying maximum-likelihood (ML) decoding for lattices in high dimensions is infeasible and forced researchers to apply other low complexity decoding methods for lattices to obtain practical capacity-achieving lattices. Integer lattices built by Construction A, D and D' can be decoded with linear complexity based on soft-decision decoding of their underlying linear binary and non-binary codes [7], [8], [9], [10], [11], [12], [13], [14]. The search for sphere-boundachieving and capacity-achieving lattices and lattice codes followed by proposing low density parity-check (LDPC) lattices [8], low density lattice codes (LDLC) [15] and integer low-density lattices based on Construction A (LDA) [9].…”