For an odd prime p and q = p^r, this paper deals with LCD codes obtained from cyclic codes of length n over a finite commutative non-chain ring \(\mathcal {R}=\mathbb {F}_{q}[u,v]/\langle u^{2}-\alpha u,v^{2}-1, uv-vu\rangle \) where α is a non-zero element in \(\mathbb {F}_{q}\). Initially, we impose certain conditions on the generator polynomials of cyclic codes when \(\gcd (n,p)=1\) and \(\gcd (n,p)\neq 1\), respectively so that these codes become LCD. Then, by defining a Gray map ψ, we show that the Gray image of an LCD code of length n over \(\mathcal {R}\) is an LCD code of length 4n over \(\mathbb {F}_{q}\). In this way, we obtain many optimal and best-known linear codes (BKLC) from the Gray images of both cyclic and LCD codes over \(\mathcal {R}\). Eventually, by applying the CSS construction on cyclic codes over \(\mathcal {R}\) that contain their Euclidean duals, we determine many superior quantum codes compared to the existing codes in the recent references.

对于奇素数p和q = p ^r，本文处理从有限可交换非链环上长度为n 的循环码获得的 LCD 码\(\mathcal {R}=\mathbb {F}_{q}[ u,v]/\langle u^{2}-\alpha u,v^{2}-1, uv-vu\rangle \)其中α是\(\mathbb {F}_{ q}\)。最初，我们分别在\(\gcd (n,p)=1\)和\(\gcd (n,p)\neq 1\)时对循环码的生成多项式施加一定的条件，使这些码成为 LCD . 然后，通过定义一个格雷图ψ，我们表明长度为n的 LCD 代码的格雷图像over \(\mathcal {R}\)是长度为 4 n的 LCD 代码，超过\(\mathbb {F}_{q}\)。通过这种方式，我们从\(\mathcal {R}\) 上的循环码和 LCD 码的格雷图像中获得了许多最佳和最著名的线性码 (BKLC )。最终，通过在包含欧几里得对偶的\(\mathcal {R}\)上的循环码上应用 CSS 构造，我们确定了与最近参考文献中的现有代码相比的许多优越的量子代码。