Galois-qudit code 

Galois-qudit Pauli strings: For a single Galois qudit, this set consists of products of \(X\)-type and \(Z\)-type operators labeled by elements \(\beta \in GF(q)\), which act on computational basis states \(|\gamma\rangle\) for \(\gamma\in GF(q)\) as \begin{align} X_{\beta}\left|\gamma\right\rangle =\left|\gamma+\beta\right\rangle \,\,\text{ and }\,\,Z_{\beta}\left|\gamma\right\rangle =e^{i\frac{2\pi}{p}\text{tr}(\beta\gamma)}\left|\gamma\right\rangle~, \tag*{(1)}\end{align} where \(\text{tr}\) is the field trace. For multiple Galois qudits, error set elements are tensor products of elements of the single-qudit error set. Tensor products of \(X\) (\(Z\)) Galois-qudit Paulis acting on different qudits are called \(X\)-type (\(Z\)-type) Galois-qudit Pauli strings. Combining the \(X\)-type and \(Z\)-type strings with a \(p\)th root of unity forms a group called the Galois-qudit Pauli group (on \(n\) Galois qudits.