Two-stage coding over the Z-channel
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In this paper, we discuss two-stage encoding algorithms capable of correcting a fraction of asymmetric errors. Suppose that we can transmit 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 moment, say n_1, it is allowed to use the complete feedback of the channel and adjust further encoding strategy based on the partial output of the channel (y_1,…,y_n_1). The goal is to transmit as much information as possible under the assumption that the total number of errors is limited by τ n, 0<τ<1. We propose an encoding strategy that uses a list-decodable code at the first stage and a high-error low-rate code at the second stage. This strategy and our converse result yield that there is a sharp transition at τ=max_0<ω<1ω + ω^3/1+4ω^3≈ 0.44 from positive rate to zero rate for two-stage encoding strategies. As side results, we derive lower and upper bounds on the size of list-decodable codes for the Z-channel and prove that for a fraction 1/4+ϵ of asymmetric errors, an error-correcting code contains at most O(ϵ^-3/2) codewords.
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