Capacity of Noisy Permutation Channels

We establish the capacity of a class communication channels introduced by Makur. The $n$ -letter input from finite alphabet is passed through discrete memoryless channel notation="LaTeX">$P_{Z|X}$ and then output sequence uniformly permuted. show that maximal rate (normalized notation="LaTeX">$\log n$ ) equals notation="LaTeX">${\frac{1}{ 2}} ( \textsf {rank}(P_{Z|X})-1)$ whenever strictly positive. This done establishing converse bound matching achievability two main ingredients our proof are: 1) sharp on Kullback-Leibler divergence sampled vector type observed DMC to an iid vector; 2) covering notation="LaTeX">$\varepsilon $ -net probability simplex with as metric. In addition positive we also find noisy permutation for notation="LaTeX">$q$ -ary erasure channels, Z-channel others.