Noisy Network Coding

A noisy network coding scheme for sending multiple sources over a general noisy network is presented. For multi-source multicast networks, the scheme naturally extends both network coding over noiseless networks by Ahlswede, Cai, Li, and Yeung, and compress-forward coding for the relay channel by Cover and El Gamal to general discrete memoryless and Gaussian networks. The scheme also recovers as special cases the results on coding for wireless relay networks and deterministic networks by Avestimehr, Diggavi, and Tse, and coding for wireless erasure networks by Dana, Gowaikar, Palanki, Hassibi, and Effros. The scheme involves message repetition coding, relay signal compression, and simultaneous decoding. Unlike previous compress--forward schemes, where independent messages are sent over multiple blocks, the same message is sent multiple times using independent codebooks as in the network coding scheme for cyclic networks. Furthermore, the relays do not use Wyner--Ziv binning as in previous compress-forward schemes, and each decoder performs simultaneous joint typicality decoding on the received signals from all the blocks without explicitly decoding the compression indices. A consequence of this new scheme is that achievability is proved simply and more generally without resorting to time expansion to extend results for acyclic networks to networks with cycles. The noisy network coding scheme is then extended to general multi-source networks by combining it with decoding techniques for interference channels. For the Gaussian multicast network, noisy network coding improves the previously established gap to the cutset bound. We also demonstrate through two popular AWGN network examples that noisy network coding can outperform conventional compress-forward, amplify-forward, and hash-forward schemes.