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

Abstract

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.