# The adaptive zero-error capacity for a class of channels with noisy feedback

## TL;DR

This paper investigates the zero-error capacity of discrete memoryless channels with noisy feedback, establishing bounds and conditions under which the capacity equals the zero-undetected error capacity of the forward channel.

## Contribution

It extends the understanding of zero-error capacity to channels with noisy feedback, providing bounds and conditions for capacity equality.

## Key findings

- Lower bound on zero-error capacity with noisy feedback equals zero-undetected error capacity
- Capacity bounds are tight under certain conditions
- Noisy feedback can be effectively characterized in terms of zero-undetected error capacity

## Abstract

The adaptive zero-error capacity of discrete memoryless channels (DMC) with noiseless feedback has been shown to be positive whenever there exists at least one channel output "disprover", i.e. a channel output that cannot be reached from at least one of the inputs. Furthermore, whenever there exists a disprover, the adaptive zero-error capacity attains the Shannon (small-error) capacity. Here, we study the zero-error capacity of a DMC when the channel feedback is noisy rather than perfect. We show that the adaptive zero-error capacity with noisy feedback is lower bounded by the forward channel's zero-undetected error capacity, and show that under certain conditions this is tight.