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Composite Matrices from Group Rings, Composite G-Codes and Constructions of Self-Dual Codes
Dougherty, Steven Gildea, Joe Korban, Adrian Kaya, Abidin
Dougherty, Steven
Gildea, Joe
Korban, Adrian
Kaya, Abidin
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2021-05-19
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Abstract
In this work, we define composite matrices which are derived from group rings. We extend the idea of G-codes to composite G-codes. We show that these codes are ideals in a group ring, where the ring is a finite commutative Frobenius ring and G is an arbitrary finite group. We prove that the dual of a composite G-code is also a composite G-code. We also define quasi-composite G-codes. Additionally, we study generator matrices, which consist of the identity matrices and the composite matrices.
Together with the generator matrices, the well known extension method, the neighbour method and its generalization, we find extremal binary self-dual codes of length 68 with new weight enumerators for the rare parameters $\gamma$ = 7; 8 and 9: In particular, we find 49 new such codes. Moreover, we show that the codes we find are inaccessible from
other constructions.
Citation
Dougherty, S. T., Gildea, J., Korban, A., & Kaya, A. (2021). Composite matrices from group rings, composite G-Codes and constructions of self-dual codes. Designs, Codes and Cryptography, 89, 1615-1638. https://doi.org/10.1007/s10623-021-00882-8
Publisher
Springer
Journal
Designs, Codes and Cryptography
Research Unit
DOI
10.1007/s10623-021-00882-8
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Article
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ISSN
0925-1022
EISSN
1573-7586