Permutation-invariant codes encoding more than one qubit

A permutation-invariant code on $m$ qubits is a subspace of the symmetric subspace of the $m$ qubits. We derive permutation-invariant codes that can encode an increasing amount of quantum information while suppressing leading-order spontaneous decay errors. To prove the result, we use elementary number theory with prior theory on permutation-invariant codes and quantum error correction.