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

Abstract

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 positive integer $n$, it is known that cyclic codes over $R$ of length $n$ are identified with ideals of the ring $R[x]/⟨x^n-1⟩$. In this paper, an explicit representation for each cyclic code over $R$ of length $n$ is provided and a formula to count the number of codewords in each code is given. Then a formula to calculate the number of cyclic codes over $R$ of length $n$ is obtained. Precisely, the dual code of each cyclic code and self-dual cyclic codes over $R$ of length $n$ are investigated. When $k=4$, some optimal quasi-cyclic codes over ${\mathbb{Z}}_4$ of length $28$ and index $4$ are obtained from cyclic codes over $R={\mathbb{Z}}_4[u]/⟨u^4⟩$.

arXiv admin note: substantial text overlap with arXiv:1511.05413