We develop an intrinsic enumerator framework for quantum error correction in unitary representations of symmetry groups. An intrinsic quantum code is a subspace of a representation $V$ of a group $G$, and errors are organized by the decomposition of the conjugation representation on $\mathcal{L}(V)$ into isotypic subspaces. Associated with any orthogonal decomposition of $\mathcal{L}(V)$ we introduce two families of quadratic enumerators, called projector and twirl enumerators, which satisfy positivity, normalization, and Knill--Laflamme type inequalities. When the conjugation representation is multiplicity--free, these enumerators are related by a linear transform that we interpret as an intrinsic MacWilliams identity. For $G=\mathrm{SU}(2)$, we compute this transform explicitly in terms of Wigner $6j$-symbols. Applied to symmetric-power representations, this gives linear programming bounds for permutation-invariant qubit and qudit codes, including extremality results for the four-qubit, seven-qubit, and three-qutrit examples treated here. We also develop the general equivariant theory in the presence of multiplicities, where the enumerators become matrix-valued, the MacWilliams transform becomes block unitary, and the resulting feasibility problem becomes semidefinite; we illustrate this theory in a first non-multiplicity-free $\mathrm{SU}(3)$ example.

翻译：针对对称群酉表示中的量子纠错，我们发展了一套内蕴枚举子框架。内蕴量子码是群$G$的表示$V$的子空间，误差通过$\mathcal{L}(V)$上共轭表示到等典型子空间的分解进行组织。对$\mathcal{L}(V)的任何正交分解，我们引入两类二次型枚举子——投影枚举子与旋转枚举子，这些枚举子满足正定性、归一化以及Knill-Laflamme型不等式。当共轭表示为无重数表示时，这些枚举子通过线性变换相关联，该变换被诠释为内蕴型MacWilliams恒等式。针对$G=\mathrm{SU}(2)$，我们以Wigner $6j$符号显式计算该变换。将其应用于对称幂表示，可得到置换不变qubit与qudit码的线性规划界，包括本文处理的四量子比特、七量子比特及三qutrit示例的极值性结果。此外，我们发展了含重数情形的一般等变理论：此时枚举子变为矩阵值函数，MacWilliams变换变为块酉矩阵，而可行问题转化为半定规划问题；我们通过首个非无重数$\mathrm{SU}(3)$实例展示了该理论。