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

Abstract

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.