  In this paper we present regular bilayer LDPC convolutional codes for
half-duplex relay channels. For the binary erasure relay channel, we prove that
the proposed code construction achieves the capacities for the source-relay
link and the source-destination link provided that the channel conditions are
known when designing the code. Meanwhile, this code enables the highest
transmission rate with decode-and-forward relaying. In addition, its regular
degree distributions can easily be computed from the channel parameters, which
significantly simplifies the code optimization. Numerical results are provided
for both binary erasure channels (BEC) and AWGN channels. In BECs, we can
observe that the gaps between the decoding thresholds and the Shannon limits
are impressively small. In AWGN channels, the bilayer LDPC convolutional code
clearly outperforms its block code counterpart in terms of bit error rate.
