On the zero-error capacity of channels with rate limited noiseless feedback

While it is known that feedback does not increase the small-error capacity of a discrete memoryless channel, noiseless feedback can increase the zero-error capacity from zero (without feedback) all the way to the small-error capacity. This result depends on the availability of noiseless output feedback, which gives the transmitter access to the exact output seen at the destination, as well as the use of variable-length codes. In this work, we consider two more realistic setups: 1) a noiseless feedback link of finite rate (which may not permit transmission of the outputs in their entirety), and 2) a noisy feedback link. We derive rates which may be achieved with zero error. Our results show that the achievable zero-error rate can vary between the zero-undetected-error capacity and the small-error capacity depending on the available feedback link rate and quality.