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New Channel Coding Lower Bounds for Noisy Permutation Channels

Lugaoze Feng, Xunan Li, Guocheng Lv, Ye jin

TL;DR

This work addresses finite-blocklength coding for noisy permutation channels where the channel is a strictly positive, full-rank square matrix combined with a random permutation. It introduces ε-packing-based divergence packing to construct a message set and derives a new nonasymptotic achievability bound that tightens prior results, complemented by a Gaussian approximation that yields a closed-form rate–blocklength tradeoff. The results are specialized to BSC and BEC permutation channels, with explicit expressions and effective numerical validation showing the new bounds outperform previous ones (e.g., Makur 2020). The combination of divergence-packing structure and Berry–Esseen-based analysis provides accurate predictions of maximal code sizes in moderate blocklengths and lays groundwork for future extension to non-full-rank or non-strictly-positive matrices.

Abstract

Motivated by the application of point-to-point communication networks and biological storage, we investigate new achievability bounds for noisy permutation channels with strictly positive and full-rank square matrices. Our new bounds use $ε$-packing with Kullback-Leibler divergence as a metric to bound the distance between distributions and are tighter than existing bounds. Additionally, Gaussian approximations of achievability bounds are derived, and the numerical evaluation shows the precision of the approximation.

New Channel Coding Lower Bounds for Noisy Permutation Channels

TL;DR

This work addresses finite-blocklength coding for noisy permutation channels where the channel is a strictly positive, full-rank square matrix combined with a random permutation. It introduces ε-packing-based divergence packing to construct a message set and derives a new nonasymptotic achievability bound that tightens prior results, complemented by a Gaussian approximation that yields a closed-form rate–blocklength tradeoff. The results are specialized to BSC and BEC permutation channels, with explicit expressions and effective numerical validation showing the new bounds outperform previous ones (e.g., Makur 2020). The combination of divergence-packing structure and Berry–Esseen-based analysis provides accurate predictions of maximal code sizes in moderate blocklengths and lays groundwork for future extension to non-full-rank or non-strictly-positive matrices.

Abstract

Motivated by the application of point-to-point communication networks and biological storage, we investigate new achievability bounds for noisy permutation channels with strictly positive and full-rank square matrices. Our new bounds use -packing with Kullback-Leibler divergence as a metric to bound the distance between distributions and are tighter than existing bounds. Additionally, Gaussian approximations of achievability bounds are derived, and the numerical evaluation shows the precision of the approximation.

Paper Structure

This paper contains 17 sections, 16 theorems, 98 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Notation
  4. System Model
  5. New Bounds on Rate
  6. Message set and Divergence Packing
  7. Achievability Bound
  8. BSC Permutation Channels
  9. BEC Permutation Channels
  10. Gaussian Approximation
  11. main result
  12. Approximation of BSC and BEC Permutation Channels
  13. Numerical Results
  14. Conclusion and Discussion
  15. Divergence Packing
  16. ...and 2 more sections

Key Result

Theorem 1

Fix a $\Delta_{|\mathcal{Y}|-1}^* \subset \Delta_{|\mathcal{Y}|-1}$ with dimension $|\mathcal{Y}|-1$. We can construct $\mathcal{N}_{r_0,|\mathcal{Y}|}^*$ by (eq:Gamma_message_set_DMCs) with $r = \sqrt{\frac{r_0}{2 \log e}}$. Then, we have where

Figures (5)

  • Figure 1: Rate-blocklength tradeoff for the BSC with crossover probability $\delta=0.11$ and average block error rate $\epsilon = 10^{-3}$: Gaussian approximation
  • Figure 2: Rate-blocklength tradeoff for the BSC with crossover probability $\delta=0.11$ and average block error rate $\epsilon = 10^{-6}$: Gaussian approximation
  • Figure 3: Rate-blocklength tradeoff for the BSC with crossover probability $\delta=0.22$ and average block error rate $\epsilon = 10^{-3}$: Gaussian approximation
  • Figure 4: Rate-blocklength tradeoff for the BSC with crossover probability $\delta=0.22$ and average block error rate $\epsilon = 10^{-6}$: Gaussian approximation
  • Figure 5: Rate-blocklength tradeoff for the BSC with crossover probability $\delta=0.11$ and average block error rate $\epsilon = 10^{-3}$:comparison of the bounds.

Theorems & Definitions (25)

  • Theorem 1
  • Proposition 1
  • Lemma 1
  • Theorem 2
  • proof
  • Theorem 3: Achievability
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • ...and 15 more