  The physical symmetries of a system play a central role in quantum error
correction. In this work we encode a qubit in a collection of systems with
angular-momentum symmetry (spins), extending the tools developed in Phys. Rev.
Lett. 127, 010504 for single large spins. By considering large spins present in
atomic systems and focusing on their collective symmetric subspace, we develop
new codes with octahedral symmetry capable of correcting errors up to second
order in angular-momentum operators. These errors include the most physically
relevant noise sources such as microwave control errors and optical pumping. We
additionally explore new qubit codes that exhibit distance scaling commensurate
with the surface code while permitting transversal single-qubit Clifford
operations.
