  Coding schemes for discrete memoryless multicast networks (DM-MN) with
rate-limited feedback from the receivers and relays to the transmitter are
proposed. The schemes improve over the noisy network coding proposed by Lim et
al.. For the single relay channel with relay-transmitter feedback, our coding
schemes recover Gabbai and Bross's results, and strictly improve on noisy
network coding, distributed decode-forward coding proposed by Lim et al., and
all known lower bounds on the achievable rates proposed for the setup without
feedback.
  The coding schemes are based on block-Markov coding, superposition coding,
sliding-window/backward decoding and hybrid relaying strategies. In our Scheme
1A, the relays and receivers use compress-forward strategy and send the
compression indices to the transmitter through the feedback pipes. After
obtaining the compression indices through feedback, the transmitter sends them
together with the source message. Each receiver uses backward decoding to
jointly decode the source message and all compression indices. Our Scheme 1B is
similar, except that here each relay not only uses compress forward to compress
its channel outputs, but also performs partial decode forward to decode a
common part of source message. Our Scheme 1C allows different relays to decode
different parts of the source message (in Scheme 1A, no relay decodes any part
of the source message; in Scheme 1B, all relays decode the same part of source
message), which can achieve higher rates than Scheme 1A and Scheme 1B.
  Motivated by the feedback coding scheme, we propose a new coding scheme for
DM-MN without feedback, where the transmitter decodes the compression indices
instead of obtaining them directly through feedback. It is shown that the
scheme still improves noisy network coding and distributed decode-forward
coding for some channels.
