Finite-Blocklength Analysis for Noisy Permutation Channels

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.