$(1-2u^3)$-constacyclic codes and quadratic residue codes over ${\mathbb{F}}_p[u]/⟨u^4-u⟩$

Abstract

Let $\mathcal{R}={\mathbb{F}}_p+u{\mathbb{F}}_p+u^2{\mathbb{F}}_p+u^3{\mathbb{F}}_p$ with $u^4=u$ be a finite non-chain ring, where $p$ is a prime congruent to $1$ modulo $3$. In this paper we study $(1-2u^3)$-constacyclic codes over the ring $\mathcal{R}$, their equivalence to cyclic codes and find their Gray images. To illustrate this, examples of $(1-2u^3)$-constacyclic codes of lengths $2^m$ for $p=7$ and of lengths $3^m$ for $p=19$ are given. We also discuss quadratic residue codes over the ring $\mathcal{R}$ and their extensions. A Gray map from $\mathcal{R}$ to ${\mathbb{F}}_p^4$ is defined which preserves self duality and gives self-dual and formally self-dual codes over ${\mathbb{F}}_p$ from extended quadratic residue codes.