Multi-way relay based deterministic broadcast with side information: Pair-wise network coding is sum-capacity optimal

We consider a K user deterministic broadcast channel where the receivers have apriori side information of a set of messages. The (deterministic) broadcast channel defined by the identity function on {0; 1}, i.e., all the receivers receive the input bit free-of-noise. The message set is inspired by the multi-way relay channel which has K users communicating through a relay, where each user has an independent message for the remaining K - 1 users. The multi-way relay channel can be converted to a broadcast channel with side information if the relay is able to decode all the messages. Thus, in the broadcast set up, there are K(K - 1) messages, with K - 1 messages intended for each receiver. Each receiver has apriori side information of a (different) set of K-1 messages. In this set up, when there are K= 2 users, it is well known that the optimal strategy for sum-capacity is to use network coding, i.e., to send an XOR of the two messages since each receiver can cancel the interfering message using its side information. The main result of this note is an outerbound for the K user version of this channel showing that scheduling users pairwise and using network coding for the scheduled pair is optimal.