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New Partial Orders of Polar Codes for BMSC

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

TL;DR

The paper addresses universal partial orders between synthesized polar channels for binary memoryless symmetric channels (BMSCs) by introducing two computable POs based on the Bhattacharyya parameter and bit-error probability. It derives new BMSC POs by leveraging known POs for the binary erasure channel (BEC) through extremal polarization inequalities, establishing bounds on $Z(W^\alpha)$ and $T(W^\alpha)=2P_e(W^\alpha)$ to transfer orders from BEC to BMSC. The authors provide concrete examples that reveal relationships not implied by channel degradation alone and determine beta-expansion bounds that preserve these POs, supported by simulations at $N=1024$ showing substantial gains in PO discovery and practical SC-decoding behavior. Overall, the work contributes a computable, physically meaningful framework for guiding polar-code construction across BMSCs and informs path-ordering strategies in decoding under realistic regimes.

Abstract

In this paper, we define partial orders (POs) of polar codes based on the Bhattacharyya parameter and the bit-error probability, respectively. These POs are applicable to arbitrary binary memoryless symmetric channel (BMSC). Leveraging the extremal inequalities of polarization transformation, we derive new POs for BMSC based on the corresponding POs observed in the Binary Erasure Channel (BEC). %Additionally, we discover more special POs in the Binary Symmetric Channel (BSC). We provide examples that demonstrate the inability of existing POs to deduce these novel POs. Furthermore, we establish upper bounds for the expansion parameter $β$ if the polar codes constructed by $β$-expansion method obey these POs.

New Partial Orders of Polar Codes for BMSC

TL;DR

The paper addresses universal partial orders between synthesized polar channels for binary memoryless symmetric channels (BMSCs) by introducing two computable POs based on the Bhattacharyya parameter and bit-error probability. It derives new BMSC POs by leveraging known POs for the binary erasure channel (BEC) through extremal polarization inequalities, establishing bounds on and to transfer orders from BEC to BMSC. The authors provide concrete examples that reveal relationships not implied by channel degradation alone and determine beta-expansion bounds that preserve these POs, supported by simulations at showing substantial gains in PO discovery and practical SC-decoding behavior. Overall, the work contributes a computable, physically meaningful framework for guiding polar-code construction across BMSCs and informs path-ordering strategies in decoding under realistic regimes.

Abstract

In this paper, we define partial orders (POs) of polar codes based on the Bhattacharyya parameter and the bit-error probability, respectively. These POs are applicable to arbitrary binary memoryless symmetric channel (BMSC). Leveraging the extremal inequalities of polarization transformation, we derive new POs for BMSC based on the corresponding POs observed in the Binary Erasure Channel (BEC). %Additionally, we discover more special POs in the Binary Symmetric Channel (BSC). We provide examples that demonstrate the inability of existing POs to deduce these novel POs. Furthermore, we establish upper bounds for the expansion parameter if the polar codes constructed by -expansion method obey these POs.
Paper Structure (17 sections, 21 theorems, 51 equations, 3 figures, 1 table)

This paper contains 17 sections, 21 theorems, 51 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Partial Orders for BMSC and BEC
  3. Channel Degradation
  4. Partial Orders in BEC
  5. New Partial Orders for BMSC
  6. POs from BEC to BMSC
  7. Examples
  8. Simulation
  9. Conclusion
  10. Proof of Proposition \ref{['BMS from BEC']}
  11. Proof of the Results for $0^m1^n$ and $1^p0^q$
  12. Proof of Proposition \ref{['general bounds for Z']}
  13. Proofs of the Equivalent Versions of Theorem \ref{['PO Z from cancel 1']}
  14. Proof of Theorem \ref{['P1']}
  15. Proofs of the Properties for $T(W)$
  16. ...and 2 more sections

Key Result

Proposition 2.1

Wu2019 If $W\preceq Q$, then

Figures (3)

  • Figure 1: The proportions of the existing POs, the newly discovered POs and the unknown pairs when $n=10$.
  • Figure 2: SC decoding performance under different $\beta$, where curves with the prefix 'MO' are the modification versions according to $P_b$.
  • Figure 3: SC decoding performance with different information sets $\mathcal{A}_{5G}$ and $\mathcal{A}_1$.

Theorems & Definitions (26)

  • Remark 1.1
  • Definition 2.1
  • Proposition 2.1
  • Proposition 2.2
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.3
  • Proposition 2.4
  • Definition 3.1
  • Proposition 3.1
  • ...and 16 more