Ideal structure of $\mathbb{Z}_q+u\mathbb{Z}_q$ and $\mathbb{Z}_q+u\mathbb{Z}_q$-cyclic codes

Abstract

In this paper, we study cyclic codes of length $n$ over $R=\mathbb{Z}_q+u\mathbb{Z}_q$, $u^2=0$, where $q$ is a power of a prime $p$ and $(n, p)=1$. We have determined the complete ideal structure of $R$. Using this,  we have obtained the structure of cyclic codes and that of their duals through the factorization of $x^n-1$ over $R$. We have also computed total number of cyclic codes of length $n$ over $R$. A necessary and sufficient condition for a cyclic code over $R$ to be self-dual is presented. We have presented a formula for total number of self-dual cyclic codes of length $n$ over $R$. A new Gray map from $R$ to $\mathbb{Z}^{2r}_p $ is defined.  Using Magma, some good cyclic codes of length $4$ over $\mathbb{Z}_9+u\mathbb{Z}_9$ are obtained.