Quantum Codes Over an Extension of \({\mathbb {Z}_4}\)

Let \(\mathfrak {A}=\mathbb Z_4+u\mathbb Z_4+v\mathbb Z_4,\) where \(u^2=u\) , \(v^2=v\) and \(uv=vu=0\) be a ring, which is an extension of \(\mathbb {Z}_{4}\) . In this article, we study the structure of cyclic codes over the ring \(\mathfrak {A}\) and define a \(\mathbb Z_2\) -linear isometry \(\Phi \) from \(\mathfrak {A}^{n}\) to \(\mathbb Z^{6n}_2.\) Based on the classical cyclic codes, we construct binary quantum codes by utilizing Gray images of cyclic codes over \(\mathfrak {A}\) . As an application, we provide some examples of binary quantum error-correcting codes.