Two-Stage Coding over the Z-Channel

A new Z-channel coding problem is addressed in this paper. Suppose that the encoder transmits n binary symbols (x(1), ..., x(n)) one-by-one over the Z-channel, in which a 1 is received if and only if a 1 is transmitted. At some designated moment, say n(1), the encoder uses noiseless feedback and adjusts further encoding strategy based on the partial output of the channel (y(1), ..., yn(1)). The goal is to transmit error-free as much information as possible under the assumption that the total number of errors inflicted by the Z-channel is limited by tau n, 0 < tau < 1. As the main contribution, we precisely characterize when exponential-sized (or positive-rate) codes exist for this model. Our proof relies on the concepts of list-decodable codes and high-error low-rate codes.