On Weight Enumeration and Structure Characterization of Polar Codes via Group Actions
Information Theory
2025-04-30 v2 math.IT
Abstract
In this article, we provide a complete characterization of codewords in polar codes with weights less than twice the minimum distance, using the group action of the lower triangular affine (LTA) group. We derive a closed-form formula for the enumeration of such codewords. Furthermore, we introduce an enhanced partial order based on weight contributions, offering refined tools for code design. Our results extend previous work on Type II codewords to a full description of Type I codewords and offer new insights into the algebraic structure underlying decreasing monomial codes, including polar and Reed-Muller codes.
Keywords
Cite
@article{arxiv.2504.19544,
title = {On Weight Enumeration and Structure Characterization of Polar Codes via Group Actions},
author = {Vlad-Florin Dragoi and Mohammad Rowshan},
journal= {arXiv preprint arXiv:2504.19544},
year = {2025}
}
Comments
A short version of this article was accepted at ISIT 2025