  We introduce a framework for constructing quantum codes defined on spheres by
recasting such codes as quantum analogues of the classical spherical codes. We
apply this framework to bosonic coding, obtaining multimode extensions of the
cat codes that can outperform previous constructions while requiring a similar
type of overhead. Our polytope-based cat codes consist of sets of points with
large separation that at the same time form averaging sets known as spherical
designs. We also recast concatenations of CSS codes with cat codes as quantum
spherical codes, revealing a new way to autonomously protect against dephasing
noise.
