Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

New Constructions of Polar Code Based on Refined Error Probability Analysis

Hassan Noghrei, Murad Abdullah

Abstract

This paper presents a refined analysis of the block error rate (BLER) of polar codes over symmetric binary-input discrete memoryless channels under successive cancellation (SC) and successive cancellation list (SCL) decoding. A novel expression for the BLER under SC decoding is derived directly in terms of the decoder's LLRs. Building on this formulation, we propose a polar code construction algorithm optimized for SC decoding and evaluate its performance under SC and dynamic SC flip (DSCF) decoding against established SC-optimized constructions, including Gaussian approximation (GA)-based and Tal-Vardy polar codes. Furthermore, by decomposing the BLER into path loss and path selection components, we derive a novel LLR-based expression for the path loss probability, which enables an SCL-optimized polar code construction method. The proposed constructions are evaluated under SCL decoding with list sizes 2, 4, and 8, and are compared with 5G standard polar codes, GA-based designs, and Reed-Muller polar codes. Simulation results show that the proposed SC-optimized polar codes achieve up to a 0.2 dB performance gain under DSCF decoding over the AWGN channel compared to benchmark constructions, and exhibit superior performance over binary symmetric channels. For SCL-optimized polar codes, the proposed method achieves comparable or improved performance across all considered list sizes, with gains of up to 0.4 dB relative to benchmark designs.

New Constructions of Polar Code Based on Refined Error Probability Analysis

Abstract

This paper presents a refined analysis of the block error rate (BLER) of polar codes over symmetric binary-input discrete memoryless channels under successive cancellation (SC) and successive cancellation list (SCL) decoding. A novel expression for the BLER under SC decoding is derived directly in terms of the decoder's LLRs. Building on this formulation, we propose a polar code construction algorithm optimized for SC decoding and evaluate its performance under SC and dynamic SC flip (DSCF) decoding against established SC-optimized constructions, including Gaussian approximation (GA)-based and Tal-Vardy polar codes. Furthermore, by decomposing the BLER into path loss and path selection components, we derive a novel LLR-based expression for the path loss probability, which enables an SCL-optimized polar code construction method. The proposed constructions are evaluated under SCL decoding with list sizes 2, 4, and 8, and are compared with 5G standard polar codes, GA-based designs, and Reed-Muller polar codes. Simulation results show that the proposed SC-optimized polar codes achieve up to a 0.2 dB performance gain under DSCF decoding over the AWGN channel compared to benchmark constructions, and exhibit superior performance over binary symmetric channels. For SCL-optimized polar codes, the proposed method achieves comparable or improved performance across all considered list sizes, with gains of up to 0.4 dB relative to benchmark designs.
Paper Structure (19 sections, 5 theorems, 68 equations, 11 figures, 3 algorithms)

This paper contains 19 sections, 5 theorems, 68 equations, 11 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. PRELIMINARIES
  3. Notations
  4. Polar Codes
  5. Error Performance of Polar Codes
  6. Block error rate Under SC decoding
  7. SC-Optimized Polar Code Construction
  8. Block error rate Under SCL decoding
  9. Path Loss Error Probability
  10. Polar Code Construction Optimized for SCL Decoding
  11. Numerical Results
  12. Performance Analysis
  13. Performance of Polar Codes Optimized for SC Decoding
  14. Performance of Polar Codes Optimized for SCL Decoding
  15. Construction Analysis
  16. ...and 4 more sections

Key Result

Lemma 1

For a polar code under genie-aided SC decoding over a symmetric B-DMC $\mathcal{W}$, the error probability $\mathcal{P}_e\!\left(u_1^i \mid y_1^N \right)$ satisfies the following symmetry property: for all $1 \le i \le N$, $(u_1^i, y_1^N) \in \mathcal{X}^i \times \mathcal{Y}^N$, and $a_1^N \in \mathcal{X}^N$, where $b_1^N = a_1^N \mathbf{G}_N$.

Figures (11)

  • Figure 2: Convergence of BLER estimation at $E_s/N_0 = 1.5$ dB for a $(256,128)$ polar code. MC (red) and the proposed approximation in (\ref{['eq:fer_bound']}) (blue) are shown versus the number of channel samples. Solid lines indicate the mean over multiple runs, shaded regions denote $\pm$ one standard deviation, and the dashed line represents a high-precision MC reference.
  • Figure 3: The BLER of the a $(256,128)$ polar code under both conventional and modified DSCF (MDSCF) decoding, evaluated with three different thresholds $\gamma \in \{0.001, 0.003, 0.005\}$.
  • Figure 4: BLER comparison of the proposed polar codes optimized for SC decoding and benchmark designs over an AWGN channel. Subfigures (a)–(c) correspond to $(N,K)=(256,128)$, $(512,256)$, and $(1024,512)$, respectively. The proposed codes are compared with GA-based and Tal-Vardy (TV) constructions under SC and DSCF decoding. For DSCF, a bit-flip order $\omega=1$ is used with up to $T=10$ attempts for $(256,128)$ and $T=8$ attempts for larger block lengths, employing 16-bit and 24-bit CRCs, respectively.
  • Figure 5: BLER comparison of the proposed polar codes optimized for SC decoding and benchmark designs over a BSC channel. Subfigure (a) shows results for $(N,K)=(256,128)$, where the proposed code is compared with GA-based and Bhattacharyya-based polar codes arikan2009channel under SC and DSCF decoding ($\omega=1$, $T=10$) using a 16-bit CRC. Subfigure (b) reports results for $(N,K)=(512,256)$, where the proposed code is compared with the same benchmarks under SC and DSCF decoding ($\omega=1$, $T=10$) using a 24-bit CRC.
  • Figure 6: BLER comparison of the proposed polar codes optimized for SCL decoding and benchmark designs over an AWGN channel. Subfigure (a) shows results for $(N,K)=(256,128)$ with a 16-bit CRC, where the proposed codes are compared with GA-based, RM-polar, and 5G polar codes under SCL decoding with $L=2,4,8$. Subfigure (b) reports results for $(N,K)=(512,256)$ with a 24-bit CRC, where the proposed codes are compared with GA-based, RM-polar, and 5G polar codes. Subfigure (c) shows results for $(N,K)=(1024,512)$ with a 24-bit CRC, where the proposed codes are compared with GA-based, RM-polar, and 5G polar codes under SCL decoding with $L=2,4,8$.
  • ...and 6 more figures

Theorems & Definitions (10)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Corollary 1
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof