Irreducible cyclic codes of length 4pn4pn and 8pn8pn

Let FqFq be a finite field of q   elements with q≡3(mod8) and p|q−1p|q−1, where p   is an odd prime. In this paper, we use matrix method to give all primitive idempotents in two rings Fq[x]/〈x4pn−1〉Fq[x]/〈x4pn−1〉 and Fq[x]/〈x8pn−1〉Fq[x]/〈x8pn−1〉. Furthermore, we obtain the dimensions and the minimum Hamming distances of all irreducible cyclic codes of length 4pn4pn and 8pn8pn over FqFq.