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Achievability Bounds on Unequal Error Protection Codes

Liuquan Yao, Shuai Yuan, Yuan Li, Huazi Zhang, Jun Wang, Guiying Yan, Zhiming Ma

TL;DR

The paper extends the Gilbert-Varshamov bound to binary unequal error protection (UEP) codes with multi-level distances and derives tighter achievability bounds for two-level protection via an intersection bound. It introduces an enlargement bound and a connected-set construction to further tighten volume estimates of unions of Hamming balls, enabling nontrivial rate gains over time-sharing under suitable conditions. Comparative analysis shows UEP can outperform time-sharing, particularly when the lower-protection level distance $d_B$ is much smaller than the higher-protection level distance $d_A$, with explicit asymptotic conditions for non-vanishing gains. Simulations corroborate the theoretical results, demonstrating gains for short codes and competitive performance for longer codes, and the work discusses extensions to $q$-ary codes and practical code constructions.

Abstract

Unequal error protection (UEP) codes can facilitate the transmission of messages with different protection levels. In this paper, we study the achievability bounds on UEP by the generalization of Gilbert-Varshamov (GV) bound. For the first time, we show that under certain conditions, UEP enhances the code rate comparing with time-sharing (TS) strategies asymptotically.

Achievability Bounds on Unequal Error Protection Codes

TL;DR

The paper extends the Gilbert-Varshamov bound to binary unequal error protection (UEP) codes with multi-level distances and derives tighter achievability bounds for two-level protection via an intersection bound. It introduces an enlargement bound and a connected-set construction to further tighten volume estimates of unions of Hamming balls, enabling nontrivial rate gains over time-sharing under suitable conditions. Comparative analysis shows UEP can outperform time-sharing, particularly when the lower-protection level distance is much smaller than the higher-protection level distance , with explicit asymptotic conditions for non-vanishing gains. Simulations corroborate the theoretical results, demonstrating gains for short codes and competitive performance for longer codes, and the work discusses extensions to -ary codes and practical code constructions.

Abstract

Unequal error protection (UEP) codes can facilitate the transmission of messages with different protection levels. In this paper, we study the achievability bounds on UEP by the generalization of Gilbert-Varshamov (GV) bound. For the first time, we show that under certain conditions, UEP enhances the code rate comparing with time-sharing (TS) strategies asymptotically.
Paper Structure (14 sections, 12 theorems, 54 equations, 4 figures, 1 table)

This paper contains 14 sections, 12 theorems, 54 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Unequal Error Protection
  3. Preliminaries
  4. Main Results
  5. GV Bound for UEP Codes
  6. Improved Bound for Two-level Protection
  7. Enlargement Union Bound
  8. Comparison between UEP and Time-sharing
  9. Time-sharing codes
  10. Theoretical comparison of UEP and TS
  11. SIMULATION
  12. Conclusion
  13. Proof of Remark \ref{['LUEP']}
  14. Proof of $g(x,y)>0,\forall x,y\in(0,1/2)$

Key Result

Lemma 2.1

There exists a binary code $C$ with length $n$, minimum distance $d$ and size $|C| \geq \frac{2^{n}}{V(n,d-1)}.$

Figures (4)

  • Figure 1: Intersection Bound
  • Figure 2: Enlargement Bound
  • Figure 3: Comparison of the Volume Bounds of the Union of Balls
  • Figure 4: Comparison on Several Bounds of TS and UEP Codes

Theorems & Definitions (20)

  • Definition 2.1
  • Remark 2.1
  • Lemma 2.1: GV bound
  • Lemma 2.2: Improved GV bound inter
  • Theorem 3.1
  • Remark 3.1
  • Corollary 3.1
  • Remark 3.2
  • Proposition 3.1
  • Definition 3.1: Connected Set
  • ...and 10 more