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A Parity-Consistent Decomposition Method for the Weight Distribution of Pre-Transformed Polar Codes

Yang Liu, Bolin Wu, Yuxin Han, Kai Niu

TL;DR

The paper tackles the computational challenge of obtaining the complete Hamming weight distribution $g_{\mathcal{C}}$ for pre-transformed polar codes. It introduces a Parity-Consistent Decomposition (PCD) framework that uses an Expanded Information Set to decouple bit dependencies and recasts the code as a union of PCD cosets, enabling recursive WD computation via convolution of leaf polar codes. A key contribution is the establishment of equivalence classes under cyclic shifts of pre-transformations, which allows selecting sparser pre-transform matrices to minimize the Expanded Information Set and reduce complexity. Empirical results show substantial reductions in the required expansion and improved efficiency for both PC-Polar and PAC codes, demonstrating the approach’s practical impact for finite-length performance prediction and design.

Abstract

This paper introduces an efficient algorithm based on the Parity-Consistent Decomposition (PCD) method to determine the WD of pre-transformed polar codes. First, to address the bit dependencies introduced by the pre-transformation matrix, we propose an iterative algorithm to construct an \emph{Expanded Information Set}. By expanding the information bits within this set into 0s and 1s, we eliminate the correlations among information bits, thereby enabling the recursive calculation of the Hamming weight distribution using the \emph{PCD method}. Second, to further reduce computational complexity, we establish the theory of equivalence classes for pre-transformed polar codes. Codes within the same equivalence class share an identical weight distribution but correspond to different \emph{Expanded Information Set} sizes. By selecting the pre-transformation matrix that minimizes the \emph{Expanded Information Set} size within an equivalence class, we optimize the computation process. Numerical results demonstrate that the proposed method significantly reduces computational complexity compared to existing deterministic algorithms.

A Parity-Consistent Decomposition Method for the Weight Distribution of Pre-Transformed Polar Codes

TL;DR

The paper tackles the computational challenge of obtaining the complete Hamming weight distribution for pre-transformed polar codes. It introduces a Parity-Consistent Decomposition (PCD) framework that uses an Expanded Information Set to decouple bit dependencies and recasts the code as a union of PCD cosets, enabling recursive WD computation via convolution of leaf polar codes. A key contribution is the establishment of equivalence classes under cyclic shifts of pre-transformations, which allows selecting sparser pre-transform matrices to minimize the Expanded Information Set and reduce complexity. Empirical results show substantial reductions in the required expansion and improved efficiency for both PC-Polar and PAC codes, demonstrating the approach’s practical impact for finite-length performance prediction and design.

Abstract

This paper introduces an efficient algorithm based on the Parity-Consistent Decomposition (PCD) method to determine the WD of pre-transformed polar codes. First, to address the bit dependencies introduced by the pre-transformation matrix, we propose an iterative algorithm to construct an \emph{Expanded Information Set}. By expanding the information bits within this set into 0s and 1s, we eliminate the correlations among information bits, thereby enabling the recursive calculation of the Hamming weight distribution using the \emph{PCD method}. Second, to further reduce computational complexity, we establish the theory of equivalence classes for pre-transformed polar codes. Codes within the same equivalence class share an identical weight distribution but correspond to different \emph{Expanded Information Set} sizes. By selecting the pre-transformation matrix that minimizes the \emph{Expanded Information Set} size within an equivalence class, we optimize the computation process. Numerical results demonstrate that the proposed method significantly reduces computational complexity compared to existing deterministic algorithms.
Paper Structure (13 sections, 6 theorems, 22 equations, 1 figure, 3 tables, 1 algorithm)

This paper contains 13 sections, 6 theorems, 22 equations, 1 figure, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries of Polar Codes
  3. Pre-Transformed Polar Codes
  4. PCD Method for Polar Codes
  5. WD Computation of Pre-Transformed Polar Codes Based on the PCD Method
  6. Determination of the Expanded Information Set
  7. WD Computation for Pre-Transformed Polar Codes
  8. Numerical Results
  9. Equivalence Classes of Pre-Transformed Polar Codes for the PCD Method
  10. Equivalence Classes of Pre-Transformed Polar Codes
  11. Optimized PCD Method via Equivalence Classes
  12. Conclusion
  13. Proof of Lemma \ref{['Lemma1']}

Key Result

Proposition 1

In the pre-transformed polar codes $\mathcal{C}$, when $u_{i}$ is a frozen bit, then $u_{j}$ needs to be expanded as 0 and 1 in the PCD method if $u_{j}$ is an information bit and $T_{j,i}=1$. Then $v_{i}$ and $v_{j}$ with $T_{j,i}=1$ are viewed as frozen bits in the PCD coset of $\mathcal{C}$.

Figures (1)

  • Figure 1: Block diagram of the proposed weight distribution computation of pre-transformed polar codes

Theorems & Definitions (18)

  • Proposition 1
  • proof
  • Remark 1
  • Definition 1
  • Definition 2
  • Theorem 1
  • proof
  • Definition 3
  • Lemma 1
  • Example 1
  • ...and 8 more