  I present a new approach for designing quantum error-correcting codes that
guarantees a physically natural implementation of Clifford operations. Inspired
by the scheme put forward by Gottesman, Kitaev, and Preskill for encoding a
qubit in an oscillator, in which Clifford operations may be performed via
Gaussian unitaries, this approach yields new schemes for encoding a qubit in a
large spin in which single-qubit Clifford operations may be performed via
spatial rotations. I construct all possible examples of such codes, provide
universal-gate-set implementations using Hamiltonians that are at most
quadratic in angular-momentum operators, and derive criteria for when these
codes exactly correct physically relevant noise channels to lowest order,
illustrating their performance numerically for specific low-dimensional
examples.
