We show that the physical consistency of magic state distillation imposes new constraints on the weight enumerators of classical error-correcting codes. We establish that for $|T\rangle$-state distillation protocols based on linear self-orthogonal $GF(4)$ codes, the distillation threshold and noise-suppression exponent are directly determined by the code's simple weight enumerator. By enforcing the physical consistency of the distillation process -- specifically, that the probability of successfully projecting onto the target state must be non-negative -- we derive a new set of constraints on classical weight enumerators. These ``quantum consistency'' constraints prove to be strictly stronger than those derived from classical invariant theory, yielding new upper bounds on the minimum distance of certain classical and quantum codes. Most notably, we show that these new constraints resolve a long-standing open problem in classical coding theory by proving the non-existence of extremal Hermitian self-dual codes over $GF(4)$ with parameters $[12m, 6m, 4m+2]$. Additionally, we use our formalism to perform an exhaustive search of distillation protocols based on linear $GF(4)$ codes with $n < 20$, finding no protocols with thresholds exceeding the 5-qubit code, and we derive analytic upper bounds on the noise-suppression exponents of such distillation routines as a function of $n$.

翻译：我们证明，魔法态蒸馏的物理一致性对经典纠错码的重量枚举子施加了新的约束。我们确立，在基于线性自正交$GF(4)$码的$|T\rangle$态蒸馏协议中，蒸馏阈值和噪声抑制指数直接由码的简单重量枚举子决定。通过强制蒸馏过程的物理一致性——具体而言，即成功投影到目标态的概率必须非负——我们推导出一组对经典重量枚举子的新约束。这些“量子一致性”约束被证明比从经典不变量理论导出的约束严格更强，从而为某些经典码和量子码的最小距离提供了新的上界。最显著的是，我们证明这些新约束通过否定参数为$[12m, 6m, 4m+2]$的极值埃尔米特自对偶$GF(4)$码的存在性，解决了经典编码理论中一个长期悬而未决的问题。此外，我们利用我们的形式体系对基于$n < 20$的线性$GF(4)$码的蒸馏协议进行了穷举搜索，未发现阈值超过5量子比特码的协议，并推导了此类蒸馏规程的噪声抑制指数作为$n$函数的解析上界。