Welcome to the Spin Kingdom.

Spin code Encodes $$K$$-dimensional Hilbert space into a $$q^n$$-dimensional ($$n$$-qudit) Hilbert space, where the canonical qudit basis consists of states of a quantum mechanical spin. In other words, canonical single-qudit states $$|^\ell_m\rangle$$ are labeled by total angular momentum $$\ell$$ (either integer or half-integer) and its $$z$$-axis projection $$m$$, with $$q=2\ell+1$$. Protection: Spin codes are often designed to protect against $$SU(2)$$ rotations by small angles. Parent of: Gross spin code. Cousins: Qubit code.
A spin code designed to realize Clifford gates using $$SU(2)$$ rotations. Codewords are subspaces of a spin's Hilbert space that house irreducible representations (irreps) of the single-qubit Clifford group (i.e., the binary octahedral ($$2O$$) subgroup of $$SU(2)$$). This is done by restricting the $$SU(2)$$ irrep to $$2O$$, and determining the carrier spaces of any nontrivial irreps of $$2O$$. Since irreps of $$2O$$ do not appear in integer spins, half-integer spins are used. Parents: Spin code.

## References

[1]
J. A. Gross, “Designing Codes around Interactions: The Case of a Spin”, Physical Review Letters 127, (2021). DOI; 2005.10910