Minimum weight codewords play a crucial role in the error correction performance of a linear block code. In this work, we establish an explicit construction for these codewords of polar codes as a sum of the generator matrix rows, which can then be used as a foundation for two applications. In the first application, we obtain a lower bound for the number of minimum-weight codewords (a.k.a. the error coefficient), which matches the exact number established previously in the literature. In the second application, we derive a novel method that modifies the information set (a.k.a. rate profile) of polar codes and PAC codes in order to reduce the error coefficient, hence improving their performance. More specifically, by analyzing the structure of minimum-weight codewords of polar codes (as special sums of the rows in the polar transform matrix), we can identify rows (corresponding to \textit{information} bits) that contribute the most to the formation of such codewords and then replace them with other rows (corresponding to \textit{frozen} bits) that bring in few minimum-weight codewords. A similar process can also be applied to PAC codes. Our approach deviates from the traditional constructions of polar codes, which mostly focus on the reliability of the sub-channels, by taking into account another important factor - the weight distribution. Extensive numerical results show that the modified codes outperform PAC codes and CRC-Polar codes at the practical block error rate of $10^{-2}$-$10^{-3}$.

翻译：最小权重码字在线性分组码的纠错性能中起着至关重要的作用。本文建立了极化码最小权重码字作为生成矩阵行之和的显式构造方法，并以此为基础开发了两项应用。在第一项应用中，我们获得了最小权重码字数（即误差系数）的下界，该下界与文献中先前确定的精确数值一致。在第二项应用中，我们提出了一种新颖的方法，通过修改极化码和PAC码的信息集（即速率配置）来降低误差系数，从而提升其性能。具体而言，通过分析极化码最小权重码字的结构（即极化变换矩阵中行的特殊和），我们可以识别出对此类码字形成贡献最大的行（对应于信息比特），并将其替换为引入最少最小权重码字的其他行（对应于冻结比特）。类似的过程也可应用于PAC码。我们的方法偏离了传统极化码主要关注子信道可靠性的构造思路，转而考虑另一个重要因素——重量分布。大量数值结果表明，在$10^{-2}$-$10^{-3}$的实际误块率范围内，改进后的码性能优于PAC码和CRC-极化码。