Capacity of Noisy Permutation Channels
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We establish the capacity of a class of communication channels introduced in [1]. The n-letter input from a finite alphabet is passed through a discrete memoryless channel P_Z|X and then the output n-letter sequence is uniformly permuted. We show that the maximal communication rate (normalized by log n) equals 1/2 (rank(P_Z|X)-1) whenever P_Z|X is strictly positive. This is done by establishing a converse bound matching the achievability of [1]. The two main ingredients of our proof are (1) a sharp bound on the entropy of a uniformly sampled vector from a type class and observed through a DMC; and (2) the covering ϵ-net of a probability simplex with Kullback-Leibler divergence as a metric. In addition to strictly positive DMC we also find the noisy permutation capacity for q-ary erasure channels, the Z-channel and others.
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