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The MacWilliams Identity for the m-Spotty Weight Enumerators over ZpRk
Juan Wang1, An Jiang2, Patrick Solé3
1Department of Base Courses, Anhui Sanlian University, Hefei 230601, China.
Entropy (Basel, Switzerland)
|January 28, 2026
Summary
This study explores m-spotty weight enumerators over mixed alphabets ZpRk, establishing the MacWilliams identity for linear codes and their duals using a novel Gray map and generalized Hadamard transform.
Area of Science:
- Coding Theory
- Abstract Algebra
- Discrete Mathematics
Background:
- The study of weight enumerators is crucial in coding theory for understanding code properties.
- Mixed alphabets like ZpRk present unique challenges in algebraic analysis.
- Previous work has focused on simpler alphabets, necessitating extensions for mixed structures.
Purpose of the Study:
- To investigate m-spotty weight enumerators over the mixed alphabet ZpRk.
- To establish the MacWilliams identity for these enumerators in the ZpRk setting.
- To introduce a generalized framework for analyzing codes over mixed alphabets.
Main Methods:
- Construction of a Gray map from Zpα×Rkβ to Zpα+kβ.
- Application of the generalized Hadamard transform.
- Utilizing the canonical additive character of Zp for algebraic analysis.
Main Results:
- Successfully constructed the Gray map for the specified mixed alphabet.
- Established the MacWilliams identity for m-spotty weight enumerators over ZpRk.
- Provided a theoretical framework applicable to linear codes and their duals over ZpRk.
Conclusions:
- The theoretical results are validated through a presented example.
- The findings extend the applicability of MacWilliams identity to more complex code structures.
- This work offers new tools for analyzing codes over mixed alphabets.
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