An information theory for erasure channels

In this paper we consider a special class of communication channels called wireless erasure channels. In these channels, symbols sent over the channel are received errorless or erased and replace by the symbols e. These channels are very relevant for modelling communication networks from the viewpoint of higher layer where applications stand. In these networks the channel appear as erasure channels where a packet arrives at destination without errors or are erased by link layer error-detection mechanism because of transmission errors or collisions. Moreover, erasures might occur because of buffer overflows caused by network congestion. Classically most of multi-user information theory researches have dealt with error channels and the erasure channel has been left out. However this class of channel is very important on different perspectives. Computer network uses erasure channels for communicating; meaning that information theory over erasure channels is relevant to these communication systems. Moreover the relative analytic simplicity of erasure channels make them attractive as an initial playground where essential characteristics of a communication problem could be extracted and the insight gained be used to tackle with the more complex error channels. In this paper we present a rapid review of known results in the context of different scenario of the erasure channel. We will also give a converse capacity bound for relay channel showing that the cut-set bound is not attainable in general by any fixed coding scheme for the single sender-single relay case. This bound shows the difference between the degraded situation where the cut-set bound is attainable and some specific case of the non-degraded situation where the cut-set bound is not attainable. The obtained reverse bound could be used for deriving tighter cut-set type bound for general multi-terminal erasure channels. However, these tighter bound will not be presented here because of lack of space.