  The multi-way relay channel (MWRC) models cooperative communication networks
in which many users exchange messages via a relay. In this paper, we consider
the finite field MWRC with correlated messages. The problem is to find all
achievable rates, defined as the number of channel uses required per reliable
exchange of message tuple. For the case of three users, we have previously
established that for a special class of source distributions, the set of all
achievable rates can be found [Ong et al., ISIT 2010]. The class is specified
by an almost balanced conditional mutual information (ABCMI) condition. In this
paper, we first generalize the ABCMI condition to the case of more than three
users. We then show that if the sources satisfy the ABCMI condition, then the
set of all achievable rates is found and can be attained using a separate
source-channel coding architecture.
