Quantum synchronizable codes are kinds of quantum error-correcting codes that
can not only correct the effects of quantum noise on qubits but also the
misalignment in block synchronization. This paper contributes to constructing
two classes of quantum synchronizable codes by the cyclotomic classes of order
two over $mathbb{Z}_{2q}$, whose synchronization capabilities can reach the
upper bound under certain conditions. Moreover, the quantum synchronizable
codes possess good error-correcting capability towards bit errors and phase
errors.

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