Cyclic codes over $$M_4 (\mathbb {F}_2+u\mathbb {F}_2)$$

Let p be a prime and $$\mathbb {F}_q$$ be a finite field for $$q=p^m$$ . In this paper, we consider the ring $$R=M_4 (\mathbb {F}_2+u\mathbb {F}_2 )$$ of $$4\times 4$$ matrices over the finite ring $$\mathbb {F}_2+u \mathbb {F}_2$$ with $$u^2=0$$ . Then R is a noncommutative non-chain ring of cardinality $$4^{16}$$ and isomorphic to the ring $$\mathbb {F}_{16}+v \mathbb {F}_{16}+v^2 \mathbb {F}_{16}+v^3 \mathbb {F}_{16}+u \mathbb {F}_{16}+uv\mathbb {F}_{16}+uv^2 \mathbb {F}_{16}+uv^3 \mathbb {F}_{16},$$ where $$v^4=0$$ , $$uv=vu$$ , $$uv^2=v^2 u$$ and $$uv^3=v^3 u$$ . Here, first we establish the structure of cyclic codes and their generators over R and later the dual (Euclidean and Hermitian both) of these cyclic codes are discussed. Further, with the help of the Gray map, we show that the image of a cyclic code is an $$\mathbb {F}_{16}$$ -linear code. Finally, we provide some non-trivial examples of linear codes with good parameters to support our derived results.