In this work, we study cyclic codes of length n over a finite commutative non-chain ring $ℛ={\mathbb{F}}_q\left[u,v\right]/〈u^2-\gamma u,v^2-ϵv,uv-vu〉$ where $\gamma ,ϵ\in {\mathbb{F}}_q^{*}$ and we find new and better quantum error-correcting codes than previously known quantum error correcting codes. Then certain constraints are imposed on the generator polynomials of cyclic codes, so these codes become linear complementary dual codes (in short LCD codes). We then verify that the Gray image of linear complementary dual codes of length n over $ℛ$ is a linear complementary dual code of length 4n over ${\mathbb{F}}_q$ by establishing a Gray map.

中文翻译：

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来自有限非链环上循环码的新量子和 LCD 码

在这项工作中，我们研究了有限可交换非链环上 长度为n的循环码$ℛ={\mathbb{F}}_q\left[你,v\right]/〈{你}^{\mathrm{2个}}-\gamma 你,v^{\mathrm{2个}}-\epsilon v,你v-v你〉$在哪里 $\gamma ,\epsilon \in {\mathbb{F}}_q^{*}$我们发现了比以前已知的量子纠错码更好的新量子纠错码。然后对循环码的生成多项式施加一定的约束，使这些码成为线性互补对偶码（简称LCD码）。然后我们验证长度为 n 的线性互补对偶码的灰度 图像$ℛ$是长度 为 4 n的线性互补对偶码${\mathbb{F}}_q$通过建立灰度图。