Motivated by the application of point-to-point communication networks and biological storage, we investigate new achievability bounds for noisy permutation channels with strictly positive and full-rank square matrices. Our new bounds use$\epsilon$-packing with Kullback-Leibler divergence as distance and are tighter than existing bounds. The key element is our analysis of the union set of error events. Additionally, Gaussian approximations of achievability bounds are derived, and the numerical evaluation shows the precision of the approximation.

Thenoisy permutation channelis a useful abstraction introduced by Makur for point-to-point communication networks and biological storage. While the asymptotic capacity results exist for this model, the characterization of the second-order asymptotics is not available. Therefore, we analyze the converse bounds for the noisy permutation channel in thefinite blocklengthregime. To do this, we present a modified minimax meta-converse for noisy permutation channels by symbol relaxation and construct an auxiliary distribution for this bound usingdivergence covering. We then show the strong converse and refined asymptotic expansions. These two conclusions hold for noisy permutation channels with strictly positive matrices (entry-wise). In addition, we obtain computable bounds for the noisy permutation channel with the binary symmetric channel (BSC), including the original computable converse bound based on the modified minimax meta-converse, the asymptotic expansion, and the$\epsilon $-capacityresult. Finally, numerical results show that the normal approximation shows remarkable precision, and our new converse bound is stronger than existing bounds.