Capacity and Zero-Error Capacity of the Chemical Channel with Feedback

We consider a family of channels, collectively referred to as the ´chemical channel´, which generalizes the trapdoor channel. We show that the feedback capacity of the chemical channel can be cast as the solution to a dynamic programming (DP) problem. We obtain numerical values for the feedback capacity of the chemical channel by approximating the solution of the DP problem using value iteration. For the special case of the trapdoor channel, by solving the DP problem analytically, we prove that the feedback capacity of the trapdoor channel is the logarithm of the golden ratio. Further, we describe a simple scheme that achieves the capacity of the trapdoor channel. The scheme has zero probability of error, which allows us to conclude that the logarithm of the golden ratio is also the zero error capacity of the chemical channel.