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Stopping Set Analysis for Concatenated Polar Code Architectures

Ziyuan Zhu, Paul H. Siegel

TL;DR

This paper presents a stopping set analysis for the factor graph of concatenated polar codes, deriving an upper bound on the size of the minimum stopping set, and shows that the exact value of the minimum stopping set can be determined with a time complexity of $O(N)$.

Abstract

This paper investigates properties of concatenated polar codes and their potential applications. We start with reviewing previous work on stopping set analysis for conventional polar codes, which we extend in this paper to concatenated architectures. Specifically, we present a stopping set analysis for the factor graph of concatenated polar codes, deriving an upper bound on the size of the minimum stopping set. To achieve this bound, we propose new bounds on the size of the minimum stopping set for conventional polar code factor graphs. The tightness of these proposed bounds is investigated empirically and analytically. We show that, in some special cases, the exact value of the minimum stopping set can be determined with a time complexity of $O(N)$, where $N$ is the codeword length. The stopping set analysis motivates a novel construction method for concatenated polar codes. This method is used to design outer polar codes for two previously proposed concatenated polar code architectures: augmented polar codes and local-global polar codes. Simulation results demonstrate the advantage of the proposed codes over previously proposed constructions based on density evolution (DE).

Stopping Set Analysis for Concatenated Polar Code Architectures

TL;DR

This paper presents a stopping set analysis for the factor graph of concatenated polar codes, deriving an upper bound on the size of the minimum stopping set, and shows that the exact value of the minimum stopping set can be determined with a time complexity of .

Abstract

This paper investigates properties of concatenated polar codes and their potential applications. We start with reviewing previous work on stopping set analysis for conventional polar codes, which we extend in this paper to concatenated architectures. Specifically, we present a stopping set analysis for the factor graph of concatenated polar codes, deriving an upper bound on the size of the minimum stopping set. To achieve this bound, we propose new bounds on the size of the minimum stopping set for conventional polar code factor graphs. The tightness of these proposed bounds is investigated empirically and analytically. We show that, in some special cases, the exact value of the minimum stopping set can be determined with a time complexity of , where is the codeword length. The stopping set analysis motivates a novel construction method for concatenated polar codes. This method is used to design outer polar codes for two previously proposed concatenated polar code architectures: augmented polar codes and local-global polar codes. Simulation results demonstrate the advantage of the proposed codes over previously proposed constructions based on density evolution (DE).

Paper Structure

This paper contains 19 sections, 14 theorems, 9 equations, 18 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Polar Codes and Systematic Polar Codes
  4. Concatenated Polar Codes
  5. Stopping Set in Factor Graph
  6. Notations from Eslami et al. Eslami2013
  7. New notations introduced in this paper
  8. Stopping set analysis for concatenated polar codes
  9. Bounds on $|MVSS(\mathcal{J})|$
  10. Lower bound on $|MVSS(\mathcal{J})|$
  11. Upper bounds on $|MVSS(\mathcal{J})|$
  12. Simulation results
  13. $|MVSS(\mathcal{J})|$ for specific choices of $\mathcal{J}$
  14. Case 1
  15. Case 2
  16. ...and 4 more sections

Key Result

Theorem 1

(Lower Bound I) Given any set $\mathcal{J}$ of information bits, we have $|MVSS(\mathcal{J})| \geq \min\limits_{j \in \mathcal{J}} f(j)$.

Figures (18)

  • Figure 1: Normal realization of the factor graph for $N=8$. An example of a SS is shown with black variable and check nodes.
  • Figure 2: The stopping tree for v(5,0) is shown with black variable and check nodes.
  • Figure 3: An example of $UT(\mathcal{J})$ for a set $\mathcal{J}$ of size 2.
  • Figure 4: Augmented structure with $N_{0}{=}4$, $N_{1} {=}8$. Orange nodes represent a SS and $\{x_4,x_5\}$ are a MVSS.
  • Figure 5: An example of the Encoding Bound, $\mathcal{J}=\{0,3,7\}$.
  • ...and 13 more figures

Theorems & Definitions (30)

  • Theorem 1
  • proof
  • Corollary 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Example 1
  • Theorem 2
  • proof
  • ...and 20 more