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Cyclicity of Binary Group Codes
Beatriz García García1, Consuelo Martínez López1, Ignacio F Rúa1
1Department of Mathematics, University of Oviedo, 33007 Oviedo, Spain.
Entropy (Basel, Switzerland)
|March 28, 2026
Summary
This study explores binary group codes, revealing that while cyclic codes are a subset of group codes, certain non-cyclic group codes possess unique structures not found in cyclic codes.
Area of Science:
- Coding Theory
- Abstract Algebra
- Computational Algebra
Background:
- Group codes are studied as ideals in group algebras.
- Cyclic codes are a well-established class within coding theory.
Purpose of the Study:
- To investigate the cyclicity of binary group codes.
- To construct and characterize self-dual group codes over specific abelian groups.
- To differentiate group codes from cyclic codes.
Main Methods:
- Identification of group codes as ideals in group algebras.
- Construction of ω|ω¯ codes over the abelian group C2×Ck.
- Utilizing MAGMA software for computational analysis of binary group codes.
Main Results:
- ω|ω¯ codes are proven to be self-dual group codes over C2×Ck.
- For even k>2, specific codes are shown to be non-permutationally equivalent to cyclic codes.
- Computational results for codes with length n<24 confirm cyclic codes are equivalent to abelian group codes.
Conclusions:
- Group codes represent a broader class than cyclic codes.
- Non-cyclic group codes exist that cannot be represented as ideals in cyclic group algebras.
- The study highlights the expanded scope of group codes beyond traditional cyclic structures.
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