Cyclic codes over ${\mathbb{Z}}_4[u]/⟨u^k⟩$ of odd length

Cao Yuan, Li Qingguo

Let $R={\mathbb{Z}}_4[u]/⟨u^k⟩={\mathbb{Z}}_4+u{\mathbb{Z}}_4+\dots +u^{k-1}{\mathbb{Z}}_4$ ($u^k=0$) where $k\in {\mathbb{Z}}^{+}$ satisfies $k\ge 2$. For any odd posi…