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.

翻译：极化码可视为递减单项式码，其丰富的代数结构由下三角仿射（LTA）群所支配。我们构建了一个通用框架来计算由单项式之和生成的码字的汉明权重，将这些权重表达为规范的二元形式，并推导出生成低权重与中等权重谱的关键结构模板（不相交和、嵌套块、互补翻转）的闭式表达式。将这些模板与LTA群作用相结合，我们获得了明确的多重度公式，从而形成了一种用于表征与枚举码字的统一代数方法。