This paper presents 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. The spin-cat code is designed to correct dominant error sources, namely processes that can be expressed as error operators linear or quadratic in the components of angular momentum. These codes, tailored to dominant error sources, can exhibit superior thresholds and lower resource overheads compared to those designed for unstructured noise models. A key component is the CNOT gate that preserves the rank of spherical tensor operators. The paper demonstrates how phase errors, analogous to phase-flip errors for qubits, can be effectively corrected. Additionally, a measurement-free error correction scheme is proposed to address amplitude errors without relying on syndrome measurements. Through an in-depth analysis of logical CNOT gate errors, it is established that the fault-tolerant threshold for error correction in the spin-cat encoding surpasses that of standard qubit-based encodings. The paper also considers a specific implementation based on neutral-atom quantum computing, with qudits encoded in the nuclear spin of $ {}^{87}Sr $, and shows 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.This paper presents 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. The spin-cat code is designed to correct dominant error sources, namely processes that can be expressed as error operators linear or quadratic in the components of angular momentum. These codes, tailored to dominant error sources, can exhibit superior thresholds and lower resource overheads compared to those designed for unstructured noise models. A key component is the CNOT gate that preserves the rank of spherical tensor operators. The paper demonstrates how phase errors, analogous to phase-flip errors for qubits, can be effectively corrected. Additionally, a measurement-free error correction scheme is proposed to address amplitude errors without relying on syndrome measurements. Through an in-depth analysis of logical CNOT gate errors, it is established that the fault-tolerant threshold for error correction in the spin-cat encoding surpasses that of standard qubit-based encodings. The paper also considers a specific implementation based on neutral-atom quantum computing, with qudits encoded in the nuclear spin of $ {}^{87}Sr $, and shows 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.