Four quantum dice are enough to protect any quantum information

Permutation-invariant quantum codes are appealing because they don't require knowing which physical qudit is which. This paper proves that exactly four physical qudits suffice to encode one logical qudit with distance-two protection for every local dimension q ≥ 2—and three is never enough. The construction reduces to a graph-coloring puzzle on the complete graph K_q, linking quantum error correction to a classical combinatorics problem with a clean, tight answer.