Let $\mathbb{ F}_{ q}$ be a finite field of cardinality $q$, $R=\mathbb{ F}_{ q}[u]/\langle u^4\rangle=\mathbb{ F}_{ q}+u\mathbb{ F}_{ q}+u^2\mathbb{ F}_{ q}+u^3\mathbb{ F}_{ q}$ $(u^4=0)$ which is a finite chain ring, and $n$ be a positive integer satisfying ${ \rm gcd}(q,n)=1$. For any $\delta,\alpha\in \mathbb{ F}_{ q}^{ \times}$, an explicit representation for all distinct $(\delta+\alpha u^2)$-constacyclic codes over $R$ of length $n$ is given, and the dual code for each of these codes is determined. For the case of $q=2^m$ and $\delta=1$, all self-dual $(1+\alpha u^2)$-constacyclic codes over $R$ of odd length $n$ are provided.
Categories: cs.IT math.IT