  In a bi-directional relay channel, two nodes wish to exchange independent
messages over a shared wireless half-duplex channel with the help of relays.
Recent work has considered information theoretic limits of the bi-directional
relay channel with a single relay. In this work we consider bi-directional
relaying with multiple relays. We derive achievable rate regions and outer
bounds for half-duplex protocols with multiple decode and forward relays and
compare these to the same protocols with amplify and forward relays in an
additive white Gaussian noise channel. We consider three novel classes of
half-duplex protocols: the (m,2) 2 phase protocol with m relays, the (m,3) 3
phase protocol with m relays, and general (m, t) Multiple Hops and Multiple
Relays (MHMR) protocols, where m is the total number of relays and 3<t< m+3 is
the number of temporal phases in the protocol. The (m,2) and (m,3) protocols
extend previous bi-directional relaying protocols for a single m=1 relay, while
the new (m,t) protocol efficiently combines multi-hop routing with
message-level network coding. Finally, we provide a comprehensive treatment of
the MHMR protocols with decode and forward relaying and amplify and forward
relaying in the Gaussian noise, obtaining their respective achievable rate
regions, outer bounds and relative performance under different SNRs and relay
geometries, including an analytical comparison on the protocols at low and high
SNR.
