New quantum codes from constacyclic codes over the ring $ R_{k,m} $

<p style='text-indent:20px;'>For any odd prime <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula>, we study constacyclic codes of length id="M3">\begin{document}$ n $\end{document}</tex-math></inline-formula> over the finite commutative non-chain ring id="M4">\begin{document}$ R_{k,m} = \mathbb{F}_{p^m}[u_1,u_2,\dots,u_k]/\langle u^2_i-1,u_iu_j-u_ju_i\rangle_{i\neq j 1,2,\dots,k} where id="M5">\begin{document}$ m,k\geq 1 are integers. We determine necessary and sufficient condition for these to contain their Euclidean duals. As an application, from dual containing codes, several MDS, new better quantum compare best known in literature obtained.</p>