Let q be a prime power and let \(f_i(w_i)\) be polynomials of degree \(n_i\) , which are not linear but split into distinct linear factors over \({\mathbb {F}}_q\) , where \(1 \le i \le k\) and \(k \ge 1\) is a positive integer. Define \(R_k\) to be the finite commutative non-chain ring \(R_k ={\mathbb {F}}_q[w_1,w_2,\ldots , w_k]/ \langle f_i(w_i), w_iw_j-w_jw_i\rangle \) . For \(\Lambda = (\lambda _0, \lambda _1, \lambda _k) \in {\mathbb {F}}_qR_1R_k\) where \(\lambda _0, \lambda _1, \lambda _k\) are units in \({\mathbb {F}}_q,R_1,R_k\) respectively, we describe constacyclic codes over \({\mathbb {F}}_qR_1R_k\) . This family of codes can be viewed as \(R_k[x]\) -submodules of \(\frac{{\mathbb {F}}_q[x]}{\langle x^{\alpha _0}-\lambda _0\rangle }\times \frac{R_1[x]}{\langle x^{\alpha _1}-\lambda _1\rangle } \times \frac{R_k[x]}{\langle x^{\alpha _k}-\lambda _k\rangle }\) . We describe the structural properties of \({\mathbb {F}}_qR_1R_k\) - \(\Lambda \) -constacyclic codes of length \((\alpha _0+\alpha _1+\alpha _k)\) and their generator polynomials. Using constacyclic codes over \({\mathbb {F}}_qR_1R_k\) , we demonstrate how to construct quantum error-correcting codes (QECC) as an application. Furthermore, we obtain new and better quantum codes with the help of a matrix Gray map.
