In this paper we study the structure of $\theta$-cyclic codes over the ring $B_k$ including its connection to quasi-$\tilde{\theta}$-cyclic codes over finite field $\mathbb{F}_{p^r}$ and skew polynomial rings over $B_k.$ We also characterize Euclidean self-dual $\theta$-cyclic codes over the rings. Finally, we give the generator polynomial for such codes and some examples of optimal Euclidean $\theta$-cyclic codes. 查看全文>>