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 = Z p m [u 1 , u 2 ,. .. , u q ]/ u 2 i = 0, u i u j = u j u i for i = j and 1 ≤ i, j ≤ q. The simplex and MacDonald codes of types α and β are obtained over R q, p,m. We characterize some linear codes over Z p m that are the torsion codes and Gray images of these simplex and MacDonald codes, and determine the minimal codes.