The MacWilliams Identity for the m-Spotty Weight Enumerators over ${\mathbb{Z}}_p{R}_k$

In this paper, we investigate the m-spotty weight enumerators over the mixed alphabet ${\mathbb{Z}}_p{R}_k$. Specifically, we construct the Gray map from ${\mathbb{Z}}_p^{\alpha }\times R_k^{\beta }$ to ${\mathbb{Z}}_p^{\alpha +k\beta }$, where $R_k={\mathbb{Z}}_p+v{\mathbb{Z}}_p+v^2{\mathbb{Z}}_p+\dots +v^{k-1}{\mathbb{Z}}_p$ with $v^k=0$ and $k\ge 5$. Based on this framework, we establish the MacWilliams identity for the m-spotty weight enumerators between a linear code and its dual over ${\mathbb{Z}}_p{R}_k$, by employing the generalized Hadamard transform and the canonical additive character of ${\mathbb{Z}}_p$. Finally, an example is presented to illustrate and validate the theoretical results.