Coding with Noiseless Feedback over the Z-channel

Abstract

In this paper, we consider encoding strategies for the Zchannel with noiseless feedback. We analyze the asymptotic case where the maximal number of errors is proportional to the blocklength, which goes to infinity. Without feedback, the asymptotic rate of error-correcting codes for the error fraction τ ≥ 1/4 is known to be zero. It was also proved that using the feedback a non-zero asymptotic rate can be achieved for the error fraction τ < 1/2. In this paper, we give an encoding strategy that achieves the asymptotic rate (1 + τ )(1 − h(τ /(1 + τ ))), which is positive for all τ < 1. Additionally, we state an upper bound on the maximal asymptotic rate of error-correcting codes for the Z-channel.