Simplex and MacDonald Codes over $R_{q}$

Abstract: In this paper, we introduce the homogeneous weight and homogeneous Gray map over the ring $$R_{q}=\mathbb{F}<em>{2}[u</em>{1},u_{2},\ldots,u_{q}]/\left\langle u_{i}<sup>{2}=0,u_{i}u_{j}=u_{j}u_{i}\right\rangle$$ for $q \geq 2$. We also consider the construction of simplex and MacDonald codes of types $\alpha$ and $\beta$ over this ring.