Construction of new quantum codes derived from constacyclic codes over $${\mathbb {F}}_{q^2}+u{\mathbb {F}}_{q^2}+\cdots +u^{r-1}{\mathbb {F}}_{q^2}$$Fq2+uFq2+⋯+ur-1Fq2

Let $$R={\mathbb {F}}_{q^2}+u{\mathbb {F}}_{q^2}+\cdots +u^{r-1}{\mathbb {F}}_{q^2}$$ be a finite non-chain ring, where q is a prime power, $$u^{r}=1$$ and $$r|(q+1)$$. In this paper, we study u-constacyclic codes over the ring R. Using the matrix of Fourier transform, a Gray map from R to $${\mathbb {F}}_{q^2}^{r}$$ is given. Under the special Gray map, we show that the image of Gray map of u-constacyclic codes over R are cyclic codes over $${\mathbb {F}}_{q^2}$$, and some new quantum codes are obtained via the Gray map and Hermitian construction from Hermitian dual-containing u-constacyclic codes.