Simplex and MacDonald codes over $$R_{q}$$ R q

Abstract

In this paper, we introduce the homogeneous weight and homogeneous Gray map over the ring \(R_{q}=\mathbb {F}_{2}[u_{1},u_{2},\ldots ,u_{q}]{/}\left\langle u_{i}^{2}=0,u_{i}u_{j}=u_{j}u_{i}\right\rangle \) for \(q \ge 1\). We also consider the construction of simplex and MacDonald codes of types \(\alpha \) and \(\beta \) over this ring and their covering radius.