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

## TL;DR

This paper derives the MacWilliams identity for m-spotty weight enumerators over a specific algebraic structure, using a Gray map and generalized Hadamard transform.

## Contribution

The paper introduces the MacWilliams identity for m-spotty weight enumerators over ZpRk using a novel Gray map and character-based methods.

## Key findings

- A Gray map from Zpα×Rkβ to Zpα+kβ is constructed for the algebraic structure Rk.
- The MacWilliams identity is established for m-spotty weight enumerators using the generalized Hadamard transform.
- An example is provided to confirm the validity of the theoretical results.

## Abstract

In this paper, we investigate the m-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 m-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.