Relaying One Bit Across a Tandem of Binary-Symmetric Channels

We consider the problem of transmitting reliably one bit of information across a tandem of binary symmetric channels interconnected by a relay/processor station. In our setting, the relay is instantaneous in the sense that its outputs are allowed to causally depend on previous received noisy bits. For this model, we investigate the optimal exponential decay rate of the average probability of error, when relaying one bit of information using n synchronous channel uses, by devising good relaying schemes.