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In this paper, it is shown that for any given single-hop communicationnetwork with two receivers, splitting messages into more than two sub-messagesin a random coding scheme is redundant. To this end, the Broadcast Channel withCognitive Relays (BCCR) is considered. A novel achievability scheme is designedfor this network. Our achievability design is derived by a systematiccombination of the best known achievability schemes for the basic buildingblocks included in the network: the Han-Kobayashi scheme for the two-userinterference channel and the Marton coding scheme for the broadcast channel.Meanwhile, in our scheme each private message is split into only twosub-messages which is identically exploited also in the Han-Kobayashi scheme.It is shown that the resultant achievable rate region includes previous resultsas well. More importantly, the procedure of the achievability design isdescribed by graphical illustrations based on directed graphs. Then, it isargued that by extending the proposed scheme on the MACCM plan of messages, onecan derive similar achievability schemes for any other single-hop communicationnetwork.

How Much Rate Splitting Is Required for a Random Coding Scheme? A new Achievable Rate Region for the Broadcast Channel with Cognitive Relays