  The multi-way relay channel is a multicast network where L users exchange
data through a relay. In this paper, the capacity region of a class of
multi-way relay channels is derived, where the channel inputs and outputs take
values over finite fields. The cut-set upper bound to the capacity region is
derived and is shown to be achievable by our proposed functional-decode-forward
coding strategy. More specifically, for the general case where the users can
transmit at possibly different rates, functional-decode-forward, combined with
rate splitting and joint source-channel decoding, is proved to achieve the
capacity region; while for the case where all users transmit at a common rate,
rate splitting and joint source-channel decoding are not required to achieve
the capacity. That the capacity-achieving coding strategies do not utilize the
users' received signals in the users' encoding functions implies that feedback
does not increase the capacity region of this class of multi-way relay
channels.
