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Outer Code Designs for Augmented and Local-Global Polar Code Architectures

Ziyuan Zhu, Paul H. Siegel

TL;DR

The paper addresses outer polar-code design for augmented and local-global concatenated polar codes under BP decoding. It introduces stopping-set construction and nonstationary density evolution (NDE) to better align outer-code reliability with inner semipolarized channels and empirical LLR distributions. The authors provide theoretical bounds, practical algorithms (including Algorithm 1 and DE-based optimization with empirical densities), and show FER gains over conventional DE and LLR-evolution methods, approaching SCL performance at moderate FERs. The work highlights the dependence of gains on the chosen connection pattern and motivates exploration of alternative outer-code designs for practical polar-code systems.

Abstract

In this paper, we introduce two novel methods to design outer polar codes for two previously proposed concatenated polar code architectures: augmented polar codes and local-global polar codes. These methods include a stopping set (SS) construction and a nonstationary density evolution (NDE) construction. Simulation results demonstrate the advantage of these methods over previously proposed constructions based on density evolution (DE) and LLR evolution.

Outer Code Designs for Augmented and Local-Global Polar Code Architectures

TL;DR

The paper addresses outer polar-code design for augmented and local-global concatenated polar codes under BP decoding. It introduces stopping-set construction and nonstationary density evolution (NDE) to better align outer-code reliability with inner semipolarized channels and empirical LLR distributions. The authors provide theoretical bounds, practical algorithms (including Algorithm 1 and DE-based optimization with empirical densities), and show FER gains over conventional DE and LLR-evolution methods, approaching SCL performance at moderate FERs. The work highlights the dependence of gains on the chosen connection pattern and motivates exploration of alternative outer-code designs for practical polar-code systems.

Abstract

In this paper, we introduce two novel methods to design outer polar codes for two previously proposed concatenated polar code architectures: augmented polar codes and local-global polar codes. These methods include a stopping set (SS) construction and a nonstationary density evolution (NDE) construction. Simulation results demonstrate the advantage of these methods over previously proposed constructions based on density evolution (DE) and LLR evolution.
Paper Structure (12 sections, 2 theorems, 4 equations, 8 figures, 3 algorithms)

This paper contains 12 sections, 2 theorems, 4 equations, 8 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Polar Codes and Systematic Polar Codes
  4. Concatenated Polar Codes
  5. Stopping Set Analysis
  6. Construction methods
  7. Stopping Set Construction
  8. Nonstationary density evolution (NDE)
  9. NDE for augmented polar codes
  10. NDE for local-global polar codes
  11. Empirical results
  12. Conclusion

Key Result

Theorem 3.1

Given any outer code position $i$ on the leftmost stage of the factor graph and its corresponding $\mathcal{J}_i$, $1\leq{i}\leq{N_{0}}$, we have $|MVSS(\mathcal{J}_i)| \geq g(G_{\mathcal{J}_i})$, where

Figures (8)

  • Figure 1: Augmented structure with $N_{0}{=}4$, $N_{1} {=}8$. Orange nodes represent a SS and $\{x_5,x_6\}$ are a MVSS.
  • Figure 2: Local-global encoder for stopping set construction.
  • Figure 3: Augmented polar code designs with $N_0{=}64$, $N_1{=}1024$.
  • Figure 4: Local decoding results for $N_0{=}256, N_1{=}N_2{=}1024$.
  • Figure 5: Global decoding results for $N_0{=}256, N_1{=}N_2{=}1024$.
  • ...and 3 more figures

Theorems & Definitions (4)

  • Theorem 3.1
  • proof
  • Example 3.2
  • Proposition 1