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

Abstract

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.