Generalized Weight Structure of Polar Codes: Selected Template Polynomials

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.