On the Weight Distribution of Weights Less than $2w_{\min}$ in Polar Codes

TL;DR

This paper extends classical results on low-weight codewords from Reed-Muller codes to decreasing polar codes, providing closed-form formulas and an efficient enumeration algorithm for codewords with weights below twice the minimum weight.

Contribution

It introduces the first closed-form expressions and polynomial-time enumeration algorithm for low-weight codewords in decreasing polar codes, expanding understanding of their weight distribution.

Findings

Closed-form expressions for codeword counts with weights less than 2w_min.

Polynomial-time enumeration algorithm for these codewords.

Extension of Reed-Muller code results to decreasing polar codes.

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.