On the Weight Spectrum of Rate-Compatible Polar Codes

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.