Quantum codes from codes over the ring \({\pmb {\mathbb {F}}}_{q}+\alpha \pmb {\mathbb {F}}_{q}\)

In this paper, we aim to obtain quantum error correcting codes from codes over a nonlocal ring \(R_q={\mathbb {F}}_q+\alpha {\mathbb {F}}_q\). We first define a Gray map \(\varphi \) from \(R_q^n\) to \({\mathbb {F}}_q^{2n}\) preserving the Hermitian orthogonality in \(R_q^n\) to both the Euclidean and trace-symplectic orthogonality in \({\mathbb {F}}_q^{2n}\). We characterize the structure of cyclic codes and their duals over \(R_q\) and derive the condition of existence for cyclic codes containing their duals over \(R_q\). By making use of the Gray map \(\varphi \), we obtain two classes of q-ary quantum codes. We also determine the structure of additive cyclic codes over \(R_{p^2}\) and give a condition for these codes to be self-orthogonal with respect to Hermitian inner product. By defining and making use of a new map \(\delta \), we construct a family of p-ary quantum codes.