On the capacity of the general trapdoor channel with feedback

The trapdoor channel is a binary input/output/state channel with state changing deterministically as the modulo2 sum of the current input, output and state. At each state, it behaves as one of two Z channels, each with crossover probability 1/2. Permuter et al. formulated the problem of finding the capacity of the trapdoor channel with feedback as a stochastic control problem. By solving the corresponding Bellman fixed-point equation, they showed that the capacity equals C = log 1+ √ 5 2 . In this paper, we consider a generalization of the trapdoor channel, whereby at each state the corresponding Z channel has crossover probability p. We characterize the capacity of this problem as the solution of a Bellman fixed-point equation corresponding to a Markov decision process (MDP). Numerical solution of this fixed-point equation reveals an unexpected behavior, that is, for a range of crossover probabilities, the capacity seems to be constant. To the authors knowledge, this is the first time that such behaviour is observed for the Shannon capacity of any channel. Our main contribution is to formalize and prove this observation. In particular, by explicitly solving the Bellman equation, we show the existence of an interval [0.5, p∗] over which the capacity remains constant.