Cyclic codes over ${\mathcal {M}}_4({\mathbb {F}}_2$)

Abstract

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.