The Gray Images of Skew-Constacyclic Codes over $\mathbb{F}_{p^m} + u\mathbb{F}_{p^m} + · · · + u^{e−1}\mathbb{F}_{p^m}$

We study the Gray images of three types of skew constacyclic codes over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+\dots+u^{e-1}\mathbb{F}_{p^m}$, where $u^e=0$. For a given automorphism $\Theta$ of $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+\dots+u^{e-1}\mathbb{F}_{p^m}$ induced by an automorphism $\theta$ of $\mathbb{F}_{p^m}$, the Gray images of $\Theta$-$(1-u^{e-1})$-constacyclic codes are shown to be $\theta$-permutation invariant codes whose algebraic structures are generalization of quasi-cyclic codes over finite fields. In addition, if the length of codes is not divisible by $p$, the Gray images of $\Theta$-cyclic and $\Theta$-$(1+u^{e-1})$-constacyclic codes are permutatively equivalent to $\theta$-permutation invariant codes. Moreover, our works generalize known results concerning the Gray images of classical cyclic, $(1-u^{e-1})$-constacyclic and $(1+u^{e-1})$-constacyclic codes over this ring.