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Quantum synchronizable codes are kinds of quantum error-correcting codes thatcan not only correct the effects of quantum noise on qubits but also themisalignment in block synchronization. This paper contributes to constructingtwo classes of quantum synchronizable codes by the cyclotomic classes of ordertwo over $\mathbb{Z}_{2q}$, whose synchronization capabilities can reach theupper bound under certain conditions. Moreover, the quantum synchronizablecodes possess good error-correcting capability towards bit errors and phaseerrors.

Quantum Synchronizable Codes From Cyclotomic Classes of Order Two over $\mathbb{Z}_{2q}$