New quantum codes and entanglement-assisted quantum codes from repeated-root cyclic codes of length \(2^rp^s\)

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 \(2^rp^s\) $2^r{p}^s$ over the finite field \(\mathbb {F}_{p^m}\) ${\mathbb{F}}_{p^m}$ . 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 identify all maximum distance separable (MDS) cyclic codes of length \(2^rp^s\) $2^r{p}^s$ , 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.