$(1-2u^k)$-constacyclic codes over ${\mathbb{F}}_p+u{\mathbb{F}}_p+u^2{\mathbb{F}}_{+}{u}^3{\mathbb{F}}_p+\cdots +u^k{\mathbb{F}}_p$

Abstract

Let ${\mathbb{F}}_p$ be a finite field and $u$ be an indeterminate. This article studies $(1-2u^k)$-constacyclic codes over the ring $\mathcal{R}={\mathbb{F}}_p+u{\mathbb{F}}_p+u^2{\mathbb{F}}_p+u^3{\mathbb{F}}_p+\cdots +u^k{\mathbb{F}}_p$ where $u^{k+1}=u$. We illustrate the generator polynomials and investigate the structural properties of these codes via decomposition theorem.