On the achievability of cut-set bound for a class of erasure relay channels

We consider an erasure relay network sender send an information flow that can be received by a receiver and a relay. The relay forward some part of the information and we assume that relay and sender transmission do not interfere with each other, however the receiver is able to receive in parallel information that are sent from sender and relay on different channels. The interferences between sender and relay transmission might be suppressed by using different sender to receiver and relay to receiver physical channels. This model is realistic for many practical scenarios in the context of wireless networks. As an example, we can give wireless mesh router networks where the relay nodes are IEEE 802.11 access points that acts as routers that route packets between different WIFI channels. The scenario is also applicable to many ad-hoc wireless networks. It is noteworthy that wireless networks appear from the viewpoint of higher layer as erasure channels; packets arrive at destination without errors or they are erased by link layer error-detection mechanisms. This important property is sometime overlooked in the information theoretical literature and it enables a lot of simplification in the analysis. We derive here a capacity region for the described channel where relay and sender activities do not interfere with each other's and the involved channels are erasure channels. The capacity is derived by defining a cut-set type bound and showing it is achievable. This bound does not make any assumption about the relay channel being degraded and use in place the entropy characterization problem solution. Moreover, a practical coding scheme is presented that could be easily implemented in real networks.