Let $S_{k}={\mathbb{F}}_{q}[u_{1},u_{2},\ldots ,u_{k}]/\langle u^{3}_{i}=u_{i},u_{i}u_{j}=u_{j}u_{i}=0 \rangle $ S k = F q [ u 1 , u 2 , … , u k ] / 〈 u i 3 = u i , u i u j = u j u i = 0 〉 , where $1\leq i,j\leq k$ 1 ≤ i , j ≤ k , $q=p^{m}$ q = p m , p is an odd prime. First, we define two new Gray maps $\phi _{k}$ ϕ k and $\varphi _{k}$ φ k , and study their Gray images. Further, we determine the structure of constacyclic codes and their dual codes, and give a necessary and sufficient conditions of constacyclic codes to contain their duals. Finally, we obtain some new quantum codes over $\mathbb{F}_{q}$ F q by using CSS construction, and compare the constructed codes better than the existing literature.

a b s t r a c tIn light of the generator polynomials of constacyclic codes over finite chain rings, the depth spectrum of constacyclic codes can be determined if (n, p) = 1.

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