Coding with Noiseless Feedback over the Z-channel

TL;DR

This paper introduces an efficient feedback encoding scheme for the Z-channel that maintains a positive rate even with high error fractions, surpassing limitations of non-feedback codes.

Contribution

It presents a new feedback-based encoding strategy for the Z-channel that achieves positive rates for any error fraction less than one, with proven upper bounds.

Findings

Feedback encoding achieves positive rate for any error fraction less than 1.

The scheme is efficient and asymptotically optimal.

Upper bounds on the rate of feedback codes are established.

Abstract

In this paper, we consider encoding strategies for the Z-channel with noiseless feedback. We analyze the combinatorial setting where the maximum number of errors inflicted by an adversary is proportional to the number of transmissions, which goes to infinity. Without feedback, it is known that the rate of optimal asymmetric-error-correcting codes for the error fraction $\tau\ge 1/4$ vanishes as the blocklength grows. In this paper, we give an efficient feedback encoding scheme with $n$ transmissions that achieves a positive rate for any fraction of errors $\tau<1$ and $n\to\infty$. Additionally, we state an upper bound on the rate of asymptotically long feedback asymmetric error-correcting codes.