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G-Codes, self-dual G-Codes and reversible G-Codes over the Ring Bj,k
Dougherty, Steven Gildea, Joe Korban, Adrian Sahinkaya, Serap
Dougherty, Steven
Gildea, Joe
Korban, Adrian
Sahinkaya, Serap
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Tarsus University; University of Chester
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2021-05-03
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Abstract
In this work, we study a new family of rings, Bj,k, whose base field is the finite field Fpr . We study the structure of this family of rings and show that each member of the family is a commutative Frobenius ring. We define a Gray map for the new family of rings, study G-codes, self-dual G-codes, and reversible G-codes over this family. In particular, we show that the projection of a G-code over Bj,k to a code over Bl,m is also a G-code and the image under the Gray map of a self-dual G-code is also a self-dual G-code when the characteristic of the base field is 2. Moreover, we show that the image of a reversible G-code under the Gray map is also a reversible G2j+k-code. The Gray images of these codes are shown to have a rich automorphism group which arises from the algebraic structure of the rings and the groups. Finally, we show that quasi-G codes, which are the images of G-codes under the Gray map, are also Gs-codes for some s.
Citation
Dougherty, S., Gildea, J., Korban, A., & Sahinkaya, S. (2021). G-Codes, self-dual G-Codes and reversible G-Codes over the Ring Bj,k. Cryptography and Communications, 13, 601–616. https://doi.org/10.1007/s12095-021-00487-x
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Springer
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Cryptography and Communications
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Article
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ISSN
1936-2447
EISSN
1936-2455