Abstract-A new coding scheme for multicasting a message over a general relay network is presented that extends both network coding for graphical networks by Ahlswede, Cai, Li, and Yeung, and partial decode-forward for relay channels by Cover and El Gamal. For the N -node Gaussian multicast network, the scheme achieves within 0.5N bits from the capacity, improving upon the best known capacity gap results. The key idea is to use multicoding at the source as in Marton coding for broadcast channels. Instead of recovering a specific part of the message as in the original partial decode-forward scheme, a relay in the proposed distributed decode-forward scheme recovers an auxiliary index that implicitly carries some information about the message and forwards it in block Markov coding. This scheme can be adapted to broadcasting multiple messages over a general relay network, extending and refining a recent result by Kannan, Raja, and Viswanath.

A new coding scheme for general N -node relay networks is presented for unicast, multicast, and broadcast.The proposed distributed decode-forward scheme combines and generalizes Marton coding for single-hop broadcast channels and the Cover-El Gamal partial decode-forward coding scheme for 3-node relay channels.The key idea of the scheme is to precode all the codewords of the entire network at the source by multicoding over multiple blocks. This encoding step allows these codewords to carry partial information of the messages implicitly without complicated rate splitting and routing. This partial information is then recovered at the relay nodes and forwarded further. For N -node Gaussian unicast, multicast, and broadcast relay networks, the scheme achieves within 0.5N bits from the cutset bound and thus from the capacity (region), regardless of the network topology, channel gains, or power constraints. Roughly speaking, distributed decode-forward is dual to noisy network coding, which generalized compress-forward to unicast, multicast, and multiple access relay networks.

A new coding scheme for broadcasting multiple messages over a general relay network is presented. The proposed distributed decode-forward scheme combines Marton coding for single-hop broadcast channels and partial decode-forward for relay channels by Cover and El Gamal. For the N -node Gaussian broadcast relay network, the scheme achieves within 0.5N bits from the capacity region, extending and refining a recent result by Kannan, Raja, and Viswanath. The main idea of the scheme is to precode all the codewords initially at the source and to decode and forward parts of them on the fly at the relays.

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