New Upper Bounds for Noisy Permutation Channels

TL;DR

This paper establishes new upper bounds for the noisy permutation channel's capacity in finite blocklength regimes, using divergence covering and modified meta-converse techniques, with implications for biological storage and communication networks.

Contribution

It introduces a modified minimax meta-converse and divergence covering methods to analyze second-order asymptotics for noisy permutation channels, providing computable bounds and improved precision.

Findings

Smaller crossover probability yields higher upper bounds at fixed blocklength.

Normal approximation closely matches the bounds, showing high accuracy.

New bounds outperform previous results in the finite blocklength regime.

Abstract

The noisy permutation channel is 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 the finite blocklength regime. To do this, we present a modified minimax meta-converse for noisy permutation channels by symbol relaxation. To derive the second-order asymptotics of the converse bound, we propose a way to use divergence covering in analysis. It enables the observation of the second-order asymptotics and the strong converse via Berry-Esseen type bounds. 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 derived from our subset covering technique, and the {\epsilon}-capacity result. We find that a smaller crossover probability provides a higher upper bound for a fixed finite blocklength, although the {\epsilon}-capacity is agnostic to the BSC parameter. Finally, numerical results show that the normal approximation shows remarkable precision, and our new converse bound is stronger than previous bounds.