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.