  We construct a fault-tolerant quantum error-correcting protocol based on a
qubit encoded in a large spin qudit using a spin-cat code, analogous to the
continuous variable cat encoding. With this, we can correct the dominant error
sources, namely processes that can be expressed as error operators that are
linear or quadratic in the components of angular momentum. Such codes tailored
to dominant error sources {can} exhibit superior thresholds and lower resource
overheads when compared to those designed for unstructured noise models. To
preserve the dominant errors during gate operations, we identify a suitable
universal gate set. A key component is the CNOT gate that preserves the rank of
spherical tensor operators. Categorizing the dominant errors as phase and
amplitude errors, we demonstrate how phase errors, analogous to phase-flip
errors for qubits, can be effectively corrected. Furthermore, we propose a
measurement-free error correction scheme to address amplitude errors without
relying on syndrome measurements. Through an in-depth analysis of logical CNOT
gate errors, we establish that the fault-tolerant threshold for error
correction in the spin-cat encoding surpasses that of standard qubit-based
encodings. We consider a specific implementation based on neutral-atom quantum
computing, with qudits encoded in the nuclear spin of $^{87}$Sr, and show how
to generate the universal gate set, including the rank-preserving CNOT gate,
using quantum control and the Rydberg blockade. These findings pave the way for
encoding a qubit in a large spin with the potential to achieve fault tolerance,
high threshold, and reduced resource overhead in quantum information
processing.
