Quadratic residue codes over the ring $\mathbb {F}_{p}[u]/\langle u^{m}-u\rangle$ and their Gray images

Abstract

Let m ≥ 2 be any natural number and let $\mathcal {R}=\mathbb {F}_{p}+u\mathbb {F}_{p}+u^{2}\mathbb {F}_{p}+\cdots +u^{m-1}\mathbb {F}_{p}$ be a finite non-chain ring, where u ^m = u and p is a prime congruent to 1 modulo (m − 1). In this paper we study quadratic residue codes over the ring $\mathcal {R}$ and their extensions. A Gray map from $\mathcal {R}^{n}$ to $(\mathbb {F}_{p}^{m})^{n}$ is defined which preserves self duality of linear codes. As a consequence, we construct self-dual, formally self-dual and self-orthogonal codes over $\mathbb {F}_{p}$. To illustrate this, several examples of self-dual, self-orthogonal and formally self-dual codes are given. Among others a [9,3,6] linear code over $\mathbb {F}_{7}$ is constructed which is self-orthogonal as well as nearly MDS. The best known linear code with these parameters (ref. Magma) is not self-orthogonal.