Quadratic residue codes over the ring 𝔽p[u]/〈 um - u 〉 and their Gray images

Let m ≥ 2 be any natural number and letR = Fp +uFp +u2Fp +· · ·+um−1Fp be a finite non-chain ring, where u = u and p is a prime congruent to 1 modulo (m − 1). In this paper we study quadratic residue codes over the ring R and their extensions. A Gray map from Rn to (Fp ) is defined which preserves self duality of linear codes. As a consequence, we construct self-dual, formally self-dual and self-orthogonal codes over Fp . 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 F7 is constructed which is selforthogonal as well as nearly MDS. The best known linear code with these parameters (ref. Magma) is not self-orthogonal.