Entanglement-assisted quantum error correcting codes (EAQECCs) play a significant role in protecting quantum information from decoherence and quantum noise. In this work, we construct six families of new EAQECCs of lengths \(n=(q^2+1)/a\), \(n=q^2+1\) and \(n=(q^2+1)/2\) from cyclic codes, where \(a=m^2+1\) (\(m\ge 1\) is odd) and q is an odd prime power with the form of \(a|(q+m)\) or \(a|(q-m)\). Moreover, those EAQECCs are entanglement-assisted quantum maximum distance separable (EAQMDS) codes when \(d\le (n+2)/2\). In particular, the length of EAQECCs we studied is more general and the method of selecting defining set is different from others. Compared with all the previously known results, the EAQECCs in this work have flexible parameters and larger minimum distance. All of these EAQECCs are new in the sense that their parameters are not covered by the quantum codes available in the literature.

纠缠辅助量子纠错码 (EAQECC) 在保护量子信息免于退相干和量子噪声方面发挥着重要作用。在这项工作中，我们构建了六个长度为\(n=(q^2+1)/a\)、\(n=q^2+1\)和\(n=(q^2+ 1)/2\)来自循环码，其中\(a=m^2+1\)（\(m\ge 1\) 是奇数），q是形式为\(a|( q+m)\)或\(a|(qm)\)。此外，当\(d\le (n+2)/2\). 尤其是我们研究的EAQECCs的长度更一般，选择定义集的方法也不同。与之前已知的所有结果相比，这项工作中的 EAQECC 具有灵活的参数和更大的最小距离。所有这些 EAQECC 都是新的，因为它们的参数没有包含在文献中可用的量子代码中。