Some Results on Density Evolution of Nonbinary SC-LDPC Ensembles Over the BEC

In this paper, we present several theoretic results on density evolution (DE) for nonbinary spatially-coupled low-density parity-check (SC-LDPC) ensembles when the transmission takes place on the binary erasure channel (BEC). In specific, we establish the duality rule for entropy for the nonbinary variable-node (VN) and check-node (CN) operators in such a scenario. We define the partial order between densities and show that the VN and CN operators exhibit the property of partial order preservation. More importantly, we explicitly construct the potential functions for uncoupled and coupled DE recursions, the forms of which almost coincide with those for binary LDPC and SC-LDPC ensembles over general binary memoryless symmetric (BMS) channels. These theoretic findings greatly facilitate the proof of threshold saturation for nonbinary SC-LDPC ensembles on the BEC. Finally, we develop the threshold saturation theorem and its converse, following the lines established by S. Kumar et al.