  Let $p$ be an odd prime and $r,s,m$ be positive integers. In this study, we
initiate our exploration by delving into the intricate structure of all
repeated-root cyclic codes and their duals with a length of $2^rp^s$ over the
finite field $\mathbb{F}_{p^m}$. Through the utilization of CSS and Steane's
constructions, a series of new quantum error-correcting (QEC) codes are
constructed with parameters distinct from all previous constructions.
Furthermore, we provide all maximum distance separable (MDS) cyclic codes of
length $2^rp^s$, which are further utilized in the construction of QEC MDS
codes. Finally, we introduce a significant number of novel
entanglement-assisted quantum error-correcting (EAQEC) codes derived from these
repeated-root cyclic codes. Notably, these newly constructed codes exhibit
parameters distinct from those of previously known constructions.
