In this paper, we construct quantum synchronizable codes (QSCs) based on the sum and intersection of cyclic codes. Further, infinite families of QSCs are obtained from BCH and duadic codes. Moreover, we show that the work of Fujiwara~cite{fujiwara1} can be generalized to repeated root cyclic codes (RRCCs) such that QSCs are always obtained, which is not the case with simple root cyclic codes. The usefulness of this extension is illustrated via examples of infinite families of QSCs from repeated root duadic codes. Finally, QSCs are constructed from the product of cyclic codes.
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