Quadratic residue codes over $\mathbb{F}_{p^r}+{u_1}\mathbb{F}_{p^r}+{u_2}\mathbb{F}_{p^r}+...+{u_t}\mathbb{F}_ {p^r}$

The purpose of this paper is to study the structure of quadratic residue codes over the ring $R=\mathbb{F}_{p^r}+u_1\mathbb{F}_{p^r}+u_2 \mathbb{F}_{p^r}+...+u_t \mathbb{F}_{p^r}$ , where $r, t ≥ 1$ and $p$ is a prime number. First, we survey known results on quadratic residue codes over the field $\mathbb{F}_{p^r}$ and give general properties with quadratic residue codes over $R$ . We introduce the Gray map from $R$ to $\mathbb{F}^{t+1}_{p^r}$ and study more details about the quadratic residue codes over the ring $R$ for $p=2, 3$ . Finally, we obtain a number of Hermitian self-dual codes over $R$ in the following two cases, where $t$ is an odd number; the first case, when $p=2$ and $r$ is an even number or $r=1$ , the second case, when $p=3$ and $r$ is an even number.