Constacyclic codes over \(\mathbb {F}_{q^{2}} \times (\mathbb {F}_{q^{2}}+v\mathbb {F}_{q^{2}})\) and their applications in constructing new quantum codes

Let \(\mathbb {F}_{q^{2}}\mathcal {R}=\mathbb {F}_{q^{2}} \times (\mathbb {F}_{q^{2}}+v\mathbb {F}_{q^{2}})\), where q is an odd prime power and \(v^{2}=v\). In this paper, we discuss the properties of linear codes and u-constacyclic codes over \(\mathbb {F}_{q^{2}}\mathcal {R}\), where \(u=(u_1,u_2)\), \(u_1\in \mathbb {F}_{q^2}^*\), \(u_2=\varepsilon (1-2v)\), and \(\varepsilon \in \mathbb {F}_{q^2}^*\). Besides, a Gray map from \(\mathbb {F}_{q^{2}}^{m}\times \mathcal {R}^{n}\) to \(\mathbb {F}_{q^{2}}^{m+2n}\) is defined, and the Gray images of linear codes and the separable \(\mathbb {F}_{q^{2}}\mathcal {R}\)-u-constacyclic codes are studied. According to the Gray images of the separable \(\mathbb {F}_{q^{2}}\mathcal {R}\)-u-constacyclic codes, some new quantum codes are obtained. Compared with the known ones, our codes have better parameters.