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 $R_k=F_{ p^m}[u]/\langle u^k\rangle =F_{ p^m}+uF_{ p^m}+\ldots+u^{ k-1}F_{ p^m}$ ($u^k=0$) and $\lambda=a_0+a_1u+\ldots+a_{ k-1}u^{ k-1}$ where $a_0, a_1,\ldots, a_{ k-1}\in F_{ p^m}$ satisfying $a_0\neq 0$ and $a_1=1$. Let $r$ be a positive integer satisfying $p^{ r-1}+1\leq k\leq p^r$. 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 $p^rN$. 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.
Categories: cs.IT math.IT