In this article, keeping the huge research prospective of the study in mind, we consider the non-commutative ring $${\mathcal {M}}_4({\mathbb {F}}_2)$$ , the set of all $$4 \times 4$$ matrices over the field $${\mathbb {F}}_2$$ and confirm that this ring is isomorphic with the ring $${\mathbb {F}}_{16}+u {\mathbb {F}}_{16}+u^2 {\mathbb {F}}_{16}+u^3{\mathbb {F}}_{16}$$ , where $$u^4=0$$ . Besides, we develop the structure of cyclic codes and their generators over the ring. Also, making use of Gray map from $${\mathcal {M}}_4({\mathbb {F}}_2)$$ to $${\mathbb {F}}_{16}^4$$ , we infer that the image of a cyclic code is a linear code. Finally, our findings are authenticated by suitable non-trivial examples.