Traditional constacyclic codes over a single finite ring can only produce quantum error-correcting codes (QECCs) with lengths constrained to integer multiples, so we investigate the constacyclic codes over the mixed alphabets \( \mathbb{F}_qR_{r} \) , where \(R_r=\mathbb{F}_q + v_{1}\mathbb{F}_q + v_{2}\mathbb{F}_q + \cdots + v_{r}\mathbb{F}_q\) with \( v_{i}^2 = v_{i}, v_{i}v_{j} = v_{j}v_{i}=0 \) , for \(1 \leq i \neq j \leq r\) and \(q\) is a prime power. Firstly, we study the linear codes over \( \mathbb{F}_qR_{r} \) and define a Gray map from \(\mathbb{F}_{q}^m \times R_{r}^n\) to \(\mathbb{F}_{q}^{m+(r+1)n}\) . Then, we establish the generator polynomials of \(\mathbb{F}_q R_r\) -t-constacyclic codes of length (m, n). In particular, we analyze their algebraic structure of separable constacyclic codes. Furthermore, we employ two methods for constructing QECCs from \( \mathbb{F}_q R_{r} \) -t-constacyclic codes: the Calderbank–Shor–Steane (CSS) construction and the Hermitian construction. These methods enable us to generate multiple QECCs with superior results.
