Simplex and MacDonald Codes over $R_q$

Abstract

In this paper, we introduce the homogeneous weight and homogeneous Gray map over the ring $R_q={\mathbb{F}}_2[u_1,u_2,\dots ,u_q]/⟨u_i^2=0,u_i{u}_j=u_j{u}_i⟩$ for $q\ge 2$. We also consider the construction of simplex and MacDonald codes of types $\alpha$ and $\beta$ over this ring.