Capacity-Achieving Codes for Noisy Feedback Channels: A Necessary Condition

In this letter, we consider a class of channels where feedback cannot increase the capacity (e.g., discrete memoryless channels). We derive a necessary condition, characterized by the directed information, on capacity-achieving channel codes for such a class of channels with noisy feedback. This condition implies that using feedback information is detrimental to the transmission rate and should be abandoned in the design of capacity-achieving channel codes.