Equivalence of constacyclic codes over finite non-chain ring and quantum codes

Let $ \mathfrak {R}_{l} = {\mathbb F}_{q}[u]/ \langle u^{l}-u \rangle $, where $ q = p^m $, $ p $ is a prime, and $ m \in \mathbf{N^+} $, $ (l-1)\mid(p-1) $. First, we study isometry and equivalence between constacyclic codes over $ \mathfrak{R}_{l} $, and we present the equivalence conditions for $ (e_1a_1+e_2a_2+\cdots+e_{l}a_{l}) $-constacyclic code and $ (e_1b_1+e_2b_2+\cdots+e_{l}b_{l}) $-constacyclic code. Further, based on the equivalence conditions, we classify constacyclic codes over $ \mathfrak{R}_{l} $. Finally, based on the equivalence of constacyclic codes over $ \mathfrak{R}_{l} $ and CSS construction, we construct some new quantum codes that are better than the existing codes in some recent references.