Let $q$ be a prime power and let $\mathcal{R}=\mathbb{F}_{q}[u_1,u_2, \cdots, u_k]/\langle f_i(u_i),u_iu_j-u_ju_i\rangle$ be a finite non-chain ring, where $f_i(u_i), 1\leq i \leq k$ are polynomials, not all linear, which split into distinct linear factors over $\mathbb{F}_{q}$. We characterize constacyclic codes over the ring $\mathcal{R}$ and study quantum codes from these. As an application, some new and better quantum codes, as compared to the best known codes, are obtained. We also prove that the choice of the polynomials $f_i(u_i),$ $1 \leq i \leq k$ is irrelevant while constructing quantum codes from constacyclic codes over $\mathcal{R}$, it depends only on their degrees. It is shown that there always exists Quantum MDS code $[[n,n-2,2]]_q$ for any $n$ with $\gcd (n,q)\neq 1.$

翻译：$qu_j_u_ju_i\rangle$ 是一个有限的非链环, 其中$f_i(u_i), 1\leq i\leq k$是多元的, 而不是全部线性, 它分为不同的线性系数, 超过$\mathbb{F ⁇ q}$。 我们在环上设定了共环代码, u_ u_ 2, u_ cdcal{R} /\ langle f_i( i_i) / u_ ju_ i\ rangle$, 是一个有限的非链性环, 其中$f_ i( u_ i), 1\leq i\ leq k$( i), 而不是全部线性系数。 我们从 $\ mathc} 上构建量子代码, 它只取决于 $\ mal_ $.n\ $( mal_ d) 。