We study permutation-invariant quantum codes in the symmetric subspace $\mathrm{Sym}^n(\mathbb{C}^q) $ of $n$ qudits of local dimension $q$. For every integer $q\geq 2$, we construct a permutation-invariant code with parameters $((4,q,2))_q$. Thus four physical qudits suffice to encode one logical qudit with distance two in the symmetric sector for every local dimension. We also show, using linear-programming constraints for permutation-invariant quantum codes, that no permutation-invariant code of dimension $q$ and distance at least $2$ exists in $\mathrm{Sym}^n(\mathbb{C}^q)$ for $n\leq 3$. Hence four qudits are necessary and sufficient. The construction has a simple representation-theoretic and combinatorial description. In the irreducible $\mathrm{SU}(q)$-module $\mathrm{Sym}^4(\mathbb{C}^q)$, the distance-two Knill-Laflamme conditions split into root and Cartan parts. By restricting supports to the even-entry occupation layer, all root-error conditions vanish automatically. The remaining Cartan conditions reduce to linear balancing constraints on packets of occupation vectors. These packets admit a natural graph-theoretic interpretation in terms of the vertices and edges of the complete graph $K_q$: for odd $q$, they are organized by the midpoint rule, while for even $q$, they are organized by a decomposition of $K_q$ into perfect matchings. In this way, the existence of minimal $((4,q,2))_q$ permutation-invariant codes is reduced to a parity-dependent edge-coloring problem on $K_q$.

翻译：我们研究局部维度为$q$的$n$个量子比特对称子空间$\mathrm{Sym}^n(\mathbb{C}^q)$中的排列不变量子码。对于每个整数$q\geq 2$，我们构造了参数为$((4,q,2))_q$的排列不变码。因此，对于任意局部维度，在对称扇区中只需四个物理量子比特即可编码一个距离为2的逻辑量子比特。我们还利用排列不变量子码的线性规划约束证明，当$n\leq 3$时，在$\mathrm{Sym}^n(\mathbb{C}^q)$中不存在维度为$q$且距离至少为2的排列不变码。因此四个量子比特是充分且必要的。该构造具有简洁的表示论和组合描述。在不可约$\mathrm{SU}(q)$模$\mathrm{Sym}^4(\mathbb{C}^q)$中，距离为2的Knill-Laflamme条件分解为根部分和Cartan部分。通过将支持限制在偶数占位层，所有根误差条件自动消失。剩余的Cartan条件简化为占位向量包上的线性平衡约束。这些包在完全图$K_q$的顶点和边方面具有自然的图论解释：对于奇数$q$，它们由中点规则组织；对于偶数$q$，它们由$K_q$分解为完美匹配的方式组织。因此，最小$((4,q,2))_q$排列不变码的存在性简化为$K_q$上与奇偶性相关的边着色问题。