Practical Encoder and Decoder for Power Constrained QC-LDPC lattices

LDPC lattices were the first family of lattices that equipped with iterative decoding algorithms under which they perform very well in high dimensions. In this paper, we introduce quasi cyclic low density parity check (QC-LDPC) lattices as a special case of LDPC lattices with one binary QC-LDPC code as their underlying code. These lattices are obtained from Construction A of lattices providing us to encode them efficiently using shift registers. To benefit from an encoder with linear complexity in dimension of the lattice, we obtain the generator matrix of these lattices in "quasi cyclic" form. We provide a low-complexity decoding algorithm of QC-LDPC lattices based on sum product algorithm. To design lattice codes, QC-LDPC lattices are combined with nested lattice shaping that uses the Voronoi region of a sublattice for code shaping. The shaping gain and shaping loss of our lattice codes with dimensions $40$, $50$ and $60$ using an optimal quantizer, are presented. Consequently, we establish a family of lattice codes that perform practically close to the sphere bound.Comment: Submitted for possible publicatio

Encoding and Indexing of Lattice Codes

Encoding and indexing of lattice codes is generalized from self-similar lattice codes to a broader class of lattices. If coding lattice $\Lambda_{\textrm{c}}$ and shaping lattice $\Lambda_{\textrm{s}}$ satisfy $\Lambda_{\textrm{s}} \subseteq \Lambda_{\textrm{c}}$, then $\Lambda_{\textrm{c}} / \Lambda_{\textrm{s}}$ is a quotient group that can be used to form a (nested) lattice code $\mathcal{C}$. Conway and Sloane's method of encoding and indexing does not apply when the lattices are not self-similar. Results are provided for two classes of lattices. (1) If $\Lambda_{\textrm{c}}$ and $\Lambda_{\textrm{s}}$ both have generator matrices in triangular form, then encoding is always possible. (2) When $\Lambda_{\textrm{c}}$ and $\Lambda_{\textrm{s}}$ are described by full generator matrices, if a solution to a linear diophantine equation exists, then encoding is possible. In addition, special cases where $\mathcal{C}$ is a cyclic code are also considered. A condition for the existence of a group homomorphism between the information and $\mathcal{C}$ is given. The results are applicable to a variety of coding lattices, including Construction A, Construction D and LDLCs. The $D_4$, $E_8$ and convolutional code lattices are shown to be good choices for the shaping lattice. Thus, a lattice code $\mathcal{C}$ can be designed by selecting $\Lambda_{\textrm{c}}$ and $\Lambda_{\textrm{s}}$ separately, avoiding competing design requirements of self-similar lattice codes