How much rate splitting is required for a random coding scheme?: A new achievable rate region for the Broadcast Channel with Cognitive Relays

In this paper, it is shown that for any given single-hop communication network with two receivers, splitting messages into more than two sub-messages in a random coding scheme is redundant. To this end, the Broadcast Channel with Cognitive Relays (BCCR) is considered. A novel achievability scheme is designed for this network. Our achievability design is derived by a systematic combination of the best known achievability schemes for the basic building blocks included in the network: the Han-Kobayashi scheme for the two-user interference channel and the Marton's coding scheme for the broadcast channel. Meanwhile, in our scheme each private message is split into only two submessages which is identically exploited also in the Han-Kobayashi scheme. It is shown that the resultant achievable rate region includes previous results as well. More importantly, the procedure of the achievability design is described by graphical illustrations based on directed graphs. Then, it is argued that by extending the proposed scheme on the MACCM plan of messages, one can derive similar achievability schemes for any other single-hop communication network.