  We derive the capacity of the binary multi-way relay channel, in which
multiple users exchange messages at a common rate through a relay. The capacity
is achieved using a novel functional-decode-forward coding strategy. In the
functional-decode-forward coding strategy, the relay decodes functions of the
users' messages without needing to decode individual messages. The functions to
be decoded by the relay are defined such that when the relay broadcasts the
functions back to the users, every user is able to decode the messages of all
other users.
