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.