Two-Stage Coding Over the Z-Channel

In this paper, we discuss two-stage encoding algorithms capable of correcting a fraction asymmetric errors. Suppose that the encoder transmits $n$ binary symbols $(x_1,\ldots,x_n)$ one-by-one over Z-channel, in which 1 is received only if transmitted. At some designated moment, say $n_1$, uses noiseless feedback and adjusts further strategy based on partial output channel $(y_1,\ldots,y_{n_1})$. The goal to transmit error-free as much information possible under assumption total number errors inflicted by Z-channel limited $\tau n$, $0<\tau<1$. We propose an list-decodable code at first stage high-error low-rate second stage. This our converse result yield there sharp transition $\tau=\max\limits_{0<w<1}\frac{w + w^3}{1+4w^3}\approx 0.44$ from positive rate zero for strategies. As side results, derive bounds size codes prove $1/4+\epsilon$ errors, error-correcting contains most $O(\epsilon^{-3/2})$ codewords.