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

Let m >= 2 be any natural number and let R = F-p + uF(p) + u(2)F(p) + center dot center dot center dot + u(m-1)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 R and their extensions. A Gray map from R-n to (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 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 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.