Low-density parity-check (LDPC) lattices were the first family of lattices to show efficient decoding in high dimensions. We consider a case of these lattices with one binary LDPC code as an underlying code. We employ encoding and decoding of the LDPC lattices in a cooperative transmission framework. We establish two efficient shaping based on hypercube and Voronoi shaping, to obtain LDPC lattice codes. Then, we propose the implementation of block Markov encoding for one-way and two-way relay networks using LDPC lattice codes. An efficient method is also required for decomposing full-rate codebook into lower rate codebooks. We apply different decomposition schemes for one-way and two-way relay channels, which are the altered versions of the decomposition methods of low density lattice code (LDLC) lattices. Due to the lower complexity of the decoding for LDPC lattices comparing with LDLCs, the complexity of our schemes is significantly lower than the ones proposed for LDLCs. The efficiency of the proposed schemes is presented using simulation results that indicate the outperforming behavior of LDLCs over LDPC lattice codes in the same dimensions. However, having lower decoding complexity enables us to increase the dimension of the lattice to compensate the existing gap between the performance of the LDPC lattice codes and the LDLCs.

Quasi-cyclic (QC) low-density parity-check (LDPC) codes which are known as QC-LDPC codes, have many applications due to their simple encoding implementation by the means of cyclic shift registers. In this paper, we construct QC-LDPC codes from group rings. A group ring is a free module (at the same time a ring) constructed in a natural way from any given ring and any given group. We present a structure based on the elements of a group ring for constructing QC-LDPC codes. Some of the previously addressed methods for constructing QC-LDPC codes based on finite fields are special cases of the proposed construction method. The constructed QC-LDPC codes perform very well over the additive white Gaussian noise channel with iterative decoding in terms of bit-error probability and block-error probability. Simulation results demonstrate that the proposed codes have competitive performance in comparison with the similar existing LDPC codes. Finally, we propose a new encoding method for the proposed group ring-based QC-LDPC codes that can be implemented faster than the current encoding methods. The encoding complexity of the proposed method is analyzed mathematically, and indicates a significate reduction in the required number of operations, even when compared to the available efficient encoding methods that have linear time and space complexities.