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

2026-01-12

AI Summary

This work addresses the high computational complexity in weight distribution analysis of pre-transformed polar codes, which arises from bit dependencies introduced by the pre-transformation matrix. To overcome this challenge, the authors propose a recursive computation method based on an extended information set and parity-check decomposition (PCD). By constructing an extended information set that eliminates correlations among information bits and leveraging equivalence class theory to select the minimal such set, the proposed approach significantly reduces computational complexity. The developed equivalence class framework offers a novel foundation for analyzing the weight distribution of pre-transformed polar codes. Numerical experiments demonstrate that the method achieves substantial gains in computational efficiency over existing deterministic algorithms while maintaining high accuracy.

Technology Category

• Machine Learning: Information Theory
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• Weight Distribution
• Pre-transformed Polar Codes
• Bit Dependencies
• Computational Complexity

Innovation

• Parity-Consistent Decomposition
• Weight Distribution
• Pre-transformed Polar Codes
• Expanded Information Set
• Equivalence Classes