Linear Codes Over the Ring $Z_8+uZ_8+vZ_8$

Abstract

In this paper, we introduce the ring $R=\mathbb{Z}_{8}+u\mathbb{Z}_{8}+v\mathbb{Z}_{8}$ where $u^{2}=u$, $v^{2}=v$, $uv=vu=0$ over which the linear codes are studied. it's shown that the ring $R=\mathbb{Z}_{8}+u\mathbb{Z}_{8}+v\mathbb{Z}_{8}$ is a commutative, characteristic 8 ring with $u^{2}=u$, $v^{2}=v$, $uv=vu=0$. Also, the ideals of $\mathbb{Z}_{8}+u\mathbb{Z}_{8}+v\mathbb{Z}_{8}$ are found. Moreover, we define the Lee distance and the Lee weight of an element of $R$ and investigate the generator matrices of the linear code and its dual.