AI 中文总结
该研究将经典与量子MacWilliams变换的本质归因于自旋运动学，通过错误分类推导其为Wigner-D旋转，揭示了码长n与旋转轴对变换的影响，打通了经典与量子编码理论的关联。
AI 中文摘要
自旋是权重枚举理论中各类MacWilliams变换背后的隐藏核心，不仅适用于量子比特、量子dit，甚至也适用于经典码。仅通过将错误划分为平凡与非平凡两类，我们从运动学角度推导出MacWilliams变换为两个标准基之间的Wigner-D旋转。在经典与量子理论中，改变码长n不会影响旋转：同一元素仅以自旋n/2的形式重新出现；而在固定n时，改变旋转轴可在各类经典与量子理论间切换。
英文摘要
Spin is the hidden engine behind the zoo of MacWilliams transforms in weight enumerator theories - not only for qubits and qudits, but even for classical codes. From nothing more than a split into trivial and nontrivial errors, we kinematically derive the MacWilliams transform as a Wigner-$D$ rotation between two canonical bases. Within each classical and quantum theory, changing the length $n$ leaves the rotation untouched: the same element simply reappears at spin $n/2$. And at fixed $n$, changing the rotation axis simply moves between the various classical and quantum theories.
