  This paper investigates the degrees of freedom (DoF) of the L-cluster, K-user
MIMO multi-way relay channel, where users in each cluster wish to exchange
messages within the cluster, and they can only communicate through the relay. A
novel DoF upper bound is derived by providing users with carefully designed
genie information. Achievable DoF is identified using signal space alignment
and multiple-access transmission. For the two-cluster MIMO multi-way relay
channel with two users in each cluster, DoF is established for the general case
when users and the relay have arbitrary number of antennas, and it is shown
that the DoF upper bound can be achieved using signal space alignment or
multiple-access transmission, or a combination of both. The result is then
generalized to the three user case. For the L-cluster K-user MIMO multi-way
relay channel in the symmetric setting, conditions under which the DoF upper
bound can be achieved are established. In addition to being shown to be tight
in a variety of scenarios of interests of the multi-way relay channel, the
newly derived upperbound also establishes the optimality of several previously
established achievable DoF results for multiuser relay channels that are
special cases of the multi-way relay channel.
