Cyclic codes over the ring GR(p^e, m)[u]/ ⟨u^k⟩

Let ℛ = GR(p^e, m)[u]/ ⟨u^k⟩ be a finite commutative ring for a prime p and any positive integers e, m and k. In this paper, we derive the explicit representation of cyclic codes over the ring ℛ of length n, where n and p are coprime. We also discuss the dual of such cyclic codes over the ring ℛ and give a sufficient condition for the codes to be self-dual. Moreover, we study quasi-cyclic codes of length kn and index k over the ring ℛ, and obtain some good codes satisfying the bound given in Dougherty and Shiromoto (2000) over the ring  as an example.