  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.
