  In this paper, we discuss two-stage encoding algorithms capable of correcting
a fraction of asymmetric errors. Suppose that the encoder transmits $n$ binary
symbols $(x_1,\ldots,x_n)$ one-by-one over the Z-channel, in which a 1 is
received 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 $(y_1,\ldots,y_{n_1})$. 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 $\tau n$,
$0<\tau<1$. 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
$\tau=\max\limits_{0<w<1}\frac{w + w^3}{1+4w^3}\approx 0.44$ from positive rate
to zero rate for two-stage encoding strategies. As side results, we derive
bounds on the size of list-decodable codes for the Z-channel and prove that for
a fraction $1/4+\epsilon$ of asymmetric errors, an error-correcting code
contains at most $O(\epsilon^{-3/2})$ codewords.
