Spatially-coupled multi-edge type LDPC codes with bounded degrees that achieve capacity on the BEC under BP decoding

Convolutional (or spatially-coupled) low-density parity-check (LDPC) codes have now been shown to approach capacity for a variety of problems. Yet, most of these results require sequences of regular LDPC ensembles with increasing variable and check degrees. Previously, Kasai and Sakaniwa showed empirically that, for the BEC, this limitation can be overcome by using spatially-coupled MacKay-Neal (MN) and Hsu-Anastasopoulos (HA) ensembles. In this paper, we prove this analytically for (k, 2, 2)-MN and (2, k, 2)-HA ensembles when k is at least 3. The proof is based on the simple approach to threshold saturation, introduced by Yedla et al., which relies on potential functions. The key step is verifying the non-negativity of a potential function associated with the uncoupled system. Along the way, we derive the potential function general multi-edge type (MET) LDPC ensembles and establish a duality relationship between dual ensembles of MET LDPC codes.