The MacWilliams Identity for the m-Spotty Weight Enumerators over ZpRk

In this paper, we investigate the <i>m</i>-spotty weight enumerators over the mixed alphabet ZpRk. Specifically, we construct the Gray map from Zpα×Rkβ to Zpα+kβ, where Rk=Zp+vZp+v2Zp+⋯+vk-1Zp with vk=0 and k≥5. Based on this framework, we establish the MacWilliams identity for the <i>m</i>-spotty weight enumerators between a linear code and its dual over ZpRk, by employing the generalized Hadamard transform and the canonical additive character of Zp. Finally, an example is presented to illustrate and validate the theoretical results.