Abstract
We study permutation-invariant quantum codes in the symmetric subspace Symⁿ(C^q) of n qudits of local dimension q. For every integer q≥ 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 Symⁿ(C^q) for n≤ 3. Hence four qudits are necessary and sufficient. The construction has a simple representation-theoretic and combinatorial description. In the irreducible SU(q)-module Sym⁴(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.