Quadratic residue codes over the ring 𝔽 p [ u ] / 〈 u m − u 〉 $\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 um = 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.