A partial decode-forward scheme for a network with N relays

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.