On the Distribution of Weights Less than 2w min in Polar Codes

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 2 w 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 2 w min . Moreover, the proposed enumeration algorithm runs in polynomial time with respect to the code length.