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.

翻译：令 $p$ 为奇素数，$r, s, m$ 为正整数。在本研究中，我们首先深入探究了有限域 $\mathbb{F}_{p^m}$ 上所有长度为 $2^rp^s$ 的重根循环码及其对偶码的复杂结构。通过利用 CSS 和 Steane 构造，我们构建了一系列参数不同于以往所有构造的新量子纠错（QEC）码。此外，我们给出了所有长度为 $2^rp^s$ 的最大距离可分（MDS）循环码，并进一步利用它们构造了 QEC MDS 码。最后，我们从这些重根循环码出发，引入了大量新颖的纠缠辅助量子纠错（EAQEC）码。值得注意的是，这些新构造的码展现出与以往已知构造不同的参数。