The Gray image of constacyclic codes over the finite chain ring $F_{p^m}[u]/⟨u^k⟩$

Abstract

Let ${\mathbb{F}}_{p^m}$ be a finite field of cardinality $p^m$, where $p$ is a prime, and $k,N$ be any positive integers. We denote $R_k=F_{p^m}[u]/⟨u^k⟩=F_{p^m}+u{F}_{p^m}+\dots +u^{k-1}{F}_{p^m}$ ($u^k=0$) and $\lambda =a_0+a_{1}u+\dots +a_{k-1}{u}^{k-1}$ where $a_0,a_1,\dots ,a_{k-1}\in F_{p^m}$ satisfying $a_0\ne 0$ and $a_1=1$. Let $r$ be a positive integer satisfying $p^{r-1}+1\le k\le p^r$. We defined a Gray map from $R_k$ to $F_{p^m}^{p^r}$ first, then prove that the Gray image of any linear $\lambda$-constacyclic code over $R_k$ of length $N$ is a distance invariant linear $a_0^{p^r}$-constacyclic code over $F_{p^m}$ of length $p^{r}N$. Furthermore, the generator polynomials for each linear $\lambda$-constacyclic code over $R_k$ of length $N$ and its Gray image are given respectively. Finally, some optimal constacyclic codes over $F_3$ and $F_5$ are constructed.