Finite-Blocklength Analysis for Noisy Permutation Channels

2026-05-25

AI Summary

This work addresses the looseness of existing finite-blocklength performance bounds for noisy permutation channels whose achievable output polytope has an affine dimension \(d\) strictly lower than that of the output simplex. By projecting the empirical distribution onto the affine hull of achievable outputs and employing Euclidean nearest-neighbor decoding, the decoding error is geometrically reduced to a one-dimensional transition event. Leveraging a refined meta-converse argument, KL divergence covering, and local binary hypothesis testing, the paper establishes the first tight finite-blocklength achievability and converse bounds that depend on the affine dimension \(d\) rather than the ambient output space dimension. The derived achievability bound hinges on local coordinate var日消息 and relative volume ratios, while the converse bound features a blocklength-dependent term of order \(d \log\sqrt{n}\), substantially improving bound tightness.

Abstract

We study finite-blocklength bounds for noisy permutation channels whose reachable output polytope may be lower-dimensional than the output simplex. Existing Gaussian achievability analyses focus on strictly positive full-rank square DMC transition matrices. The capacity result for arbitrary strictly positive DMCs is established through a weak converse, while available strong converse bounds in the lower-dimensional setting can scale with the dimension of the output simplex rather than with that of the reachable output polytope. On the achievability side, messages are placed on a simplex lattice in affine coordinates, and decoding is performed by projecting the empirical output distribution onto the reachable affine hull followed by Euclidean nearest-neighbor decoding. Writing $d$ for the affine dimension of the reachable output polytope, a geometric reduction converts decoding errors into $d(d+1)$ one-dimensional transfer events, yielding a refined Gaussian achievability lower bound based on averaged local coordinate variances and a relative volume ratio. On the converse side, a modified meta-converse, a Kullback--Leibler divergence covering, and a local binary-testing bound yield a strong converse whose blocklength-dependent term is $d\log\sqrt n$, up to a bounded additive remainder.

Problem

• noisy permutation channels
• finite-blocklength analysis
• reachable output polytope
• strong converse
• achievable bounds

Innovation

• finite-blocklength
• noisy permutation channels
• reachable output polytope
• Gaussian achievability
• strong converse