Irregular low-density parity-check lattices

We construct lattices with high coding gains based on nested low-density parity-check (LDPC) codes by Construction D´.We generalize the LDPC lattices [1] to have irregular degrees and call them irregular LDPC lattices. To construct good irregular LDPC lattices, we optimize the degree distributions by using density evolution and a modified sequential quadratic programming (SQP) and show our optimized irregular LDPC lattice has a threshold 0.48 dB from the capacity, which is about 0.6 dB better than the regular one in [1]. To construct a Tanner graph corresponding to LDPC lattices, we generalize the progressive-edge growth (PEG) algorithm and show a lower bound on the girth of the graph. Simulation is performed using the sum-product algorithm. Also, we compare the performance of joint decoding and multi-stage decoding [2] and show the advantage of joint decoding.