On the Weight Spectrum of Rate-Compatible Polar Codes

2024-10-25

arXiv.org

AI Summary

This work investigates the weight spectrum characteristics of rate-compatible polar codes—including punctured, shortened, and bit-reversal-truncated variants—with emphasis on efficiently computing the number of minimum-weight codewords and the average weight spectrum. Method: We establish the first systematic analytical framework for the weight spectrum of rate-compatible polar codes; develop polynomial-time algorithms that overcome the exponential complexity barrier of conventional methods, enabling efficient computation of the average weight spectrum under random upper-triangular pre-transforms for punctured and shortened codes; and perform the first complete enumeration of minimum-weight codewords for three prominent short-code constructions: quasi-uniform puncturing, Wang–Liu truncation, and bit-reversal truncation. Results: Simulation results validate the performance gains from spectrum optimization, demonstrating improved error-correction capability. The proposed framework provides a high-accuracy, low-complexity theoretical foundation and practical tools for code design in 5G-Advanced and 6G short-packet communications.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Combinatorial Optimization

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• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

The weight spectrum plays a crucial role in the performance of error-correcting codes. Despite substantial theoretical exploration into polar codes with mother code length, a framework for the weight spectrum of rate-compatible polar codes remains elusive. In this paper, we address this gap by enumerating the number of minimum-weight codewords for quasi-uniform punctured, Wang-Liu shortened, and bit-reversal shortened polar codes. Additionally, we propose efficient algorithms for computing the average spectrum of random upper-triangular pre-transformed shortened and punctured polar codes. Notably, our algorithms operate with polynomial complexity relative to the code length. Simulation results affirm that our findings can substantially enhance the practical construction of rate-compatible polar codes, and leading to an improved weight spectrum.

Problem

• Polar Codes
• Weight Distribution
• Fast Computation

Innovation

• Adjustable Rate Polar Codes
• Weight Distribution
• Efficient Algorithm