Generalized Weight Structure of Polar Codes: Selected Template Polynomials

TL;DR

This paper introduces a comprehensive algebraic framework for analyzing the weight structure of polar codes, leveraging template polynomials and group actions to explicitly characterize codeword weights and multiplicities.

Contribution

It develops a unified method using template polynomials and the LTA group to compute and enumerate codeword weights in polar codes, revealing their algebraic structure.

Findings

Closed-form expressions for codeword weights

Explicit multiplicity formulas for codeword enumeration

A unified algebraic framework for polar code analysis

Abstract

Polar codes can be viewed as decreasing monomial codes, revealing a rich algebraic structure governed by the lower-triangular affine (LTA) group. We develop a general framework to compute the Hamming weight of codewords generated by sums of monomials, express these weights in a canonical dyadic form, and derive closed expressions for key structural templates (disjoint sums, nested blocks, complementary flips) that generate the low and intermediate weight spectrum. Combining these templates with the LTA group action, we obtain explicit multiplicity formulas, yielding a unified algebraic method to characterize and enumerate codewords.