On the Distribution of Weights Less Than 2win Polar Codes

Zicheng Ye, Yuan Li, Huazi Zhang, Jun Wang, Guiying Yan, Zhiming Ma

Abstract: 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.