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In this paper, we study linear codes over ring  R k   =   p m   [ u  1 ,  u  2 ,⋯, u k  ]/〈u   i    2$\begin{array}{} u^{2}_{i} \end{array} $    =  u i  ,  u i   u j   =  u j   u i  〉 where  k  ≥ 1 and 1 ≤  i ,  j  ≤  k . We define a Gray map fromR   k   ntoFp  m2  k   n$\begin{array}{} R_{k}^n\,\,\text{to}\,\,{\mathbb F}_{p^m}^{2^kn} \end{array} $    and give the generator polynomials of constacyclic codes over  R k  . We also study the MacWilliams identities of linear codes over  R k  .

Constacyclic codes over 𝔽 p m [ u 1 , u 2 ,⋯, u k ]/〈 u i 2 = u i , u i u j = u j u i 〉