The number of low-weight codewords is critical to the performance of error-correcting codes. In 1970, Kasami and Tokura characterized the codewords of Reed-Muller (RM) codes whose weights are less than $2w_{\min}$, where $w_{\min}$ represents the minimum weight. In this paper, we extend their results to decreasing polar codes. We present the closed-form expressions for the number of codewords in decreasing polar codes with weights less than $2w_{\min}$. Moreover, the proposed enumeration algorithm runs in polynomial time with respect to the code length.

翻译：低重量码字的数量对纠错码的性能至关重要。1970年，Kasami和Tokura刻画了Reed-Muller（RM）码中重量小于$2w_{\min}$的码字，其中$w_{\min}$表示最小重量。本文将其结果推广至递减极化码。我们给出了递减极化码中重量小于$2w_{\min}$的码字数目的闭式表达式。此外，所提出的枚举算法在码长上具有多项式时间复杂度。