  We study a discrete-memoryless relay network consisting of one source, one
destination and N relays, and design a scheme based on partial decode-forward
relaying. The source splits its message into one common and N+1 private parts,
one intended for each relay. It encodes these message parts using Nth-order
block Markov coding, in which each private message part is independently
superimposed on the common parts of the current and N previous blocks. Using
simultaneous sliding window decoding, each relay fully recovers the common
message and its intended private message with the same block index, then
forwards them to the following nodes in the next block. This scheme can be
applied to any network topology. We derive its achievable rate in a compact
form. The result reduces to a known decode-forward lower bound for an N-relay
network and partial decode-forward lower bound for a two-level relay network.
We then apply the scheme to a Gaussian two-level relay network and obtain its
capacity lower bound considering power constraints at the transmitting nodes.
