A Tutorial on Weight Structure of Polar Codes
arXiv:2608.18493 · doi:10.1109/MBITS.2026.3693264
Abstract
This tutorial introduces the algebraic foundations underlying the weight structure of polar codes. Using a monomial-based polynomial formalism, we explain how polar and Reed-Muller codes can be viewed as decreasing monomial codes, enabling systematic characterization and enumeration of low-weight codewords. Its goal is to provide an accessible introduction to affine automorphisms, orbit-based descriptions of minimum and low-weight codewords, and their role in weight enumeration. Through illustrative examples and high-level overviews of recursive and coset-based techniques, the paper aims to prepare readers for deeper engagement with the recent technical literature on the weight distribution of polar codes.
This paper has been accepted by IEEE BITS: The Information Theory Magazine and will appear in the upcoming issue. 14 pages, 7 figures, 2 tables