On the Listsize Capacity with Feedback

The listsize capacity of a discrete memoryless channel is the largest transmission rate for which the expectation---or, more generally, the $\rho$-th moment---of the number of messages that could have produced the output of the channel approaches one as the blocklength tends to infinity. We show that for channels with feedback this rate is upper-bounded by the maximum of Gallager's $E_0$ function divided by $\rho$, and that equality holds when the zero-error capacity of the channel is positive. To establish this inequality we prove that feedback does not increase the cutoff rate. Relationships to other notions of channel capacity are explored.