The MacWilliams Identity for the m-Spotty Weight Enumerators over ℤpRk
Juan Wang, An Jiang, Patrick Solé
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.
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.
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Taxonomy
TopicsCoding theory and cryptography · graph theory and CDMA systems · Error Correcting Code Techniques