Lattice codes are elegant and powerful structures that not only can achieve the capacity of the AWGN channel but also are a key ingredient to many multiterminal schemes that exploit linearity properties. However, constructing lattice codes that can realize these benefits with low complexity is still a challenging problem. In this paper, efficient encoding and decoding algorithms are proposed for multilevel binary LDPC lattices constructed via Construction D' whose complexity is linear in the total number of coded bits. Moreover, a generalization of Construction D' is proposed that relaxes some of the nesting constraints on the component codes, leading to a simpler and improved design. Based on this construction, low-complexity multilevel LDPC lattices are designed whose performance under multistage lattice decoding is comparable to that of polar lattices on the power-unconstrained AWGN channel and close to that of conventional (non-lattice) coded modulation schemes on the power-constrained AWGN channel.
@cite_38: A simple sphere bound gives the best possible tradeoff between the volume per point of an infinite array L and its error probability on an additive white Gaussian noise (AWGN) channel. It is shown that the sphere bound can be approached by a large class of coset codes or multilevel coset codes with multistage decoding, including certain binary lattices. These codes have structure of the kind that has been found to be useful in practice. Capacity curves and design guidance for practical codes are given. Exponential error bounds for coset codes are developed, generalizing Poltyrev's (1994) bounds for lattices. These results are based on the channel coding theorems of information theory, rather than the Minkowski-Hlawka theorem of lattice theory.
@cite_23: 
@cite_12: In this paper, we propose a new class of lattices constructed from polar codes, namely polar lattices, to achieve the capacity @math of the additive white Gaussian-noise (AWGN) channel for any signal-to-noise ratio (SNR). Our construction follows the multilevel approach of , where on each level we construct a capacity-achieving polar code. The component polar codes are naturally nested, thereby fulfilling the requirement of the multilevel lattice construction. We prove that polar lattices are AWGN-good, in the sense that the error probability (for infinite lattice decoding) vanishes for any fixed volume-to-noise ratio (VNR) greater than @math . Furthermore, using the technique of source polarization, we propose discrete Gaussian shaping over the polar lattice to satisfy the power constraint. The proposed polar lattices permit low-complexity multistage successive cancellation decoding. Both the construction and shaping are explicit, and the overall complexity of encoding and decoding is @math for any fixed target error probability.
@cite_11: In this work we introduce and establish the concept of turbo lattices. We employ a routine method for constructing lattices, called Construction D, to construct turbo lattices. This kind of construction needs a set of nested linear codes as its underlying structure. We benefit from turbo codes as our bases codes. Therefore, we first build a set of nested turbo codes based on nested interleavers and nested convolutional codes. Then by means of these codes, along with construction D, we construct turbo lattices. Several properties of Construction D lattices and specially many characteristics of turbo lattices including the minimum distance, coding gain, kissing number and the probability of error under a maximum likelihood decoder over AWGN channel are investigated.
Multilevel lattice constructions based on binary codes are potentially harder to design, but have the promise of complexity that scales linearly with the number of levels. Moreover, they are known to be AWGN-good under multistage decoding @cite_38 . Construction D has been used in @cite_11 to produce turbo lattices and in @cite_12 to construct polar lattices; the latter are shown to be AWGN-good with encoding and decoding complexity @math . Construction D has also been used in @cite_23 to construct spatially-coupled LDPC lattices, which were shown to be AWGN-good under multistage belief propagation decoding. However, both the encoding and the cancellation step that has to be performed at each decoding stage rely on the generator matrices of the component LDPC codes, which are generally dense, leading to an overall high complexity.