Coding Theorems for Noisy Permutation Channels

TL;DR

This paper introduces an information theoretic framework for noisy permutation channels, establishing capacity bounds and efficient coding schemes for a class of channels where outputs are permuted, with implications for network communication and DNA storage.

Contribution

It defines the capacity of noisy permutation channels, provides bounds for DMCs, and introduces a simple, capacity-achieving coding scheme for certain channels.

Findings

Lower bound on capacity based on the rank of the DMC's stochastic matrix

Upper bounds for strictly positive DMCs

Capacity-achieving coding scheme for full rank DMCs

Abstract

In this paper, we formally define and analyze the class of noisy permutation channels. The noisy permutation channel model constitutes a standard discrete memoryless channel (DMC) followed by an independent random permutation that reorders the output codeword of the DMC. While coding theoretic aspects of this model have been studied extensively, particularly in the context of reliable communication in network settings where packets undergo transpositions, and closely related models of DNA based storage systems have also been analyzed recently, we initiate an information theoretic study of this model by defining an appropriate notion of noisy permutation channel capacity. Specifically, on the achievability front, we prove a lower bound on the noisy permutation channel capacity of any DMC in terms of the rank of the stochastic matrix of the DMC. On the converse front, we establish two upper bounds on the noisy permutation channel capacity of any DMC whose stochastic matrix is strictly positive (entry-wise). Together, these bounds yield coding theorems that characterize the noisy permutation channel capacities of every strictly positive and "full rank" DMC, and our achievability proof yields a conceptually simple, computationally efficient, and capacity achieving coding scheme for such DMCs. Furthermore, we also demonstrate the relation between the output degradation preorder over channels and noisy permutation channel capacity. In fact, the proof of one of our converse bounds exploits a degradation result that constructs a symmetric channel for any DMC such that the DMC is a degraded version of the symmetric channel. Finally, we illustrate some examples such as the special cases of binary symmetric channels and (general) erasure channels. Somewhat surprisingly, our results suggest that noisy permutation channel capacities are generally quite agnostic to the parameters that define the DMCs.