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

Abstract

Low density parity check (LDPC) lattices were the first family of lattices equipped with iterative decoding algorithms. We introduce quasi-cyclic LDPC (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 the Construction A of lattices providing us to encode them efficiently using shift registers. To benefit from an encoder with linear complexity in the lattice dimension, we obtain the generator matrix of these lattices in quasi-cyclic form. We generalize the proposed quasi-cyclic form of the generator matrix for other Construction A lattices, namely the LDA lattices, with a non-binary QC-LDPC code as their underlying code. We provide a low-complexity decoding algorithm of QC LDPC-lattices based on the sum product algorithm. To design lattice codes, QC LDPC-lattices are combined with the nested lattice shaping that uses the Voronoi region of a sublattice for shaping. The shaping gain and the shaping loss of our lattice codes with dimensions 40, 50, and 60 using an optimal quantizer, are presented. The guidelines for applying efficient shaping methods, like hypercube shaping, for QC LDPC-lattices are also given. Consequently, we establish a family of lattice codes that perform practically close to the sphere bound.