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New Upper Bounds for Noisy Permutation Channels

Lugaoze Feng, Baoji Wang, Guocheng Lv, Xvnan Li, Luhua Wang, Ye jin

TL;DR

This work analyzes the converse bounds for the noisy permutation channel in the finite blocklength regime and obtains computable bounds for the noisy permutation channel with the binary symmetric channel (BSC), including the original computable converse bound.

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 ε-capacity result. We find that a smaller crossover probability provides a higher upper bound for a fixed finite blocklength, although the ε-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.

New Upper Bounds for Noisy Permutation Channels

TL;DR

This work analyzes the converse bounds for the noisy permutation channel in the finite blocklength regime and obtains computable bounds for the noisy permutation channel with the binary symmetric channel (BSC), including the original computable converse bound.

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 ε-capacity result. We find that a smaller crossover probability provides a higher upper bound for a fixed finite blocklength, although the ε-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.
Paper Structure (22 sections, 10 theorems, 106 equations, 7 figures)

This paper contains 22 sections, 10 theorems, 106 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. System Model and Preliminaries
  4. Noisy Permutation Channels
  5. Preliminaries
  6. Codes for Noisy Permutation Channels
  7. Converse Bounds for Noisy Permutation Channels
  8. A Modified Minimax Meta-Converse
  9. Asymptotics and Bounds Based on Divergence Covering
  10. Main Results
  11. Explicit Bounds for BSC Permutation Channels
  12. Numerical Results
  13. Precision of the Normal Approximation
  14. Properties of the Normal Approximation
  15. Comparison with Previous Converse Bounds
  16. ...and 7 more sections

Key Result

Proposition 1

For the noisy permutation channel with strictly positive matrices, let $W$ is any channel from $\mathcal{X}$ to $\mathcal{Y}$ and $\mathcal{N}_r$ is any set of divergence covering centers, we have where we set $Q_{Y^n}=\hat{Q}_{Y^n}$.

Figures (7)

  • Figure 1: Illustration of a communication system with a random permutation followed by a DMC
  • Figure 2: Comparison of the converse bound and its approximation for the BSC permutation channel with crossover probability $\delta = 0.11$ and average error probability $\epsilon = 10^{-3}$.
  • Figure 3: Comparison of the converse bound and its approximation for the BSC permutation channel with crossover probability $\delta = 0.11$ and average error probability $\epsilon = 10^{-4}$.
  • Figure 4: Comparison of the converse bound and its approximation for the BSC permutation channel with crossover probability $\delta = 0.22$ and average error probability $\epsilon = 10^{-3}$.
  • Figure 5: Comparison of the converse bound and its approximation for the BSC permutation channel with crossover probability $\delta = 0.22$ and average error probability $\epsilon = 10^{-4}$.
  • ...and 2 more figures

Theorems & Definitions (22)

  • Proposition 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 1
  • Remark 1
  • Corollary 1
  • proof
  • Remark 2
  • Theorem 2
  • ...and 12 more