New classes of codes over $$R_{q,p,m}={\mathbb {Z}}_{p^{m}}[u_{1}, u_{2}, \ldots , u_{ q}]/\left\langle u_{i}^{2}=0,u_{i}u_{j}=u_{j}u_{i}\right\rangle $$ and their applications

Abstract

In this paper, we consider the construction of new classes of linear codes over the ring $$R_{q,p,m}={\mathbb {Z}}_{p^{m}}[u_{1}, u_{2}, \ldots , u_{ q}]/\left\langle u_{i}^{2}=0,u_{i}u_{j}=u_{j}u_{i}\right\rangle $$ for $$i\ne j$$ and $$1 \le i,j \le q$$ . The simplex and MacDonald codes of types $$\alpha $$ and $$\beta $$ are obtained over $$R_{q,p,m}$$ . We characterize some linear codes over $${\mathbb {Z}}_{p^{m}}$$ that are the torsion codes and Gray images of these simplex and MacDonald codes, and determine the minimal codes.