Codebook information does not reduce output entropy when rate is above capacity

Under the standard random coding framework, the conditional entropy rate of the channel output given the codebook information is investigated. It is shown that there exists an interesting dichotomy: with respect to the unconditional entropy rate of the output, this conditional entropy rate is reduced when the communication rate is below the channel capacity, but remains the same when the communication rate is above the channel capacity. Hence, the channel capacity plays a role of the critical point. Shannon used the random codebook argument to prove that any communication rate below the channel capacity is achievable, i.e., the destination can successfully decode the transmitted codeword based on the codebook used at the source. However, this cannot be done when the communication rate is above the channel capacity. In this case, our evaluation shows that the codebook information cannot be used to reduce the output entropy rate, and instead, the output entropy rate turns to be the same as if a completely random, instead of a specific codebook, was used at the source. This characterization of the conditional entropy rate sheds some light on the optimal design of the compress-and-forward relay schemes, where a long standing question is whether the source´s codebook information can be used for more effective compressions at the relay. In this paper, specifically, we show that if lossless compression is to be performed, then the codebook information cannot be used to reduce the compression rate at the relay.