The weight distribution of error correction codes is a critical determinant of their error-correcting performance, making enumeration of utmost importance. In the case of polar codes, the minimum weight $\wm$ (which is equal to minimum distance $d$) is the only weight for which an explicit enumerator formula is currently available. Having closed-form weight enumerators for polar codewords with weights greater than the minimum weight not only simplifies the enumeration process but also provides valuable insights towards constructing better polar-like codes. In this paper, we contribute towards understanding the algebraic structure underlying higher weights by analyzing Minkowski sums of orbits. Our approach builds upon the lower triangular affine (LTA) group of decreasing monomial codes. Specifically, we propose a closed-form expression for the enumeration of codewords with weight $1.5\wm$. Our simulations demonstrate the potential for extending this method to higher weights.

翻译：错误校正码的重量分布是其纠错性能的关键决定因素，因此重量枚举至关重要。对于极化码，当前仅显式枚举器公式适用于最小重量$\wm$（等于最小距离$d$）。获取大于最小重量的极化码字闭式重量枚举器，不仅简化了枚举过程，还为构建更好的类极化码提供了宝贵见解。本文通过分析轨道明可夫斯基和，致力于理解高权重背后的代数结构。我们的方法基于递减单项码的下三角仿射群，具体而言，提出了重量为$1.5\wm$的码字枚举闭式表达式。仿真结果表明，该方法具有扩展至高重量的潜力。