On the Weight Spectrum of Rate-Compatible Polar Codes

TL;DR

This paper develops a theoretical framework and efficient algorithms to analyze the weight spectrum of rate-compatible polar codes, enhancing understanding of their performance and enabling better code design.

Contribution

It introduces the first comprehensive analysis of the weight spectrum for various rate-compatible polar codes and proposes polynomial-time algorithms for spectrum estimation.

Findings

Accurate estimation of the weight spectrum improves performance prediction.

Algorithms operate with polynomial complexity, making analysis feasible for large codes.

Simulation results validate the theoretical and algorithmic approaches.

Abstract

The weight spectrum plays a crucial role in the performance of error-correcting codes. Despite substantial theoretical exploration of 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 presenting the theoretical results for enumerating the number of minimum-weight codewords for quasi-uniform punctured, Wang-Liu shortened, and bit-reversal shortened decreasing 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 yield a precise estimation of the performance of rate-compatible polar codes.