Generalized Weight Structure of Polar Codes: Selected Template Polynomials

Mohammad Rowshan; Vlad-Florin Dragoi

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arXiv · 2601.10362

Generalized Weight Structure of Polar Codes: Selected Template Polynomials

Mohammad Rowshan·Vlad-Florin Dragoi

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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.

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Mohammad Rowshan, Vlad-Florin Dragoi. 2026-01-15. Generalized Weight Structure of Polar Codes: Selected Template Polynomials. https://arxiv.org/abs/2601.10362

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