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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. Toillustrate 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 quadraticresidue 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 andgives self-dual and formally self-dual codes over $\mathbb{F}_{p}$ fromextended quadratic residue codes.

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