(1−2u ³)-constacyclic codes and quadratic residue codes over $\mathbb {F}_{p}[u]/\langle u^{4}-u\rangle$

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.