A Tutorial on Weight Structure of Polar Codes

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.