G-codes, self-dual G-codes and reversible G-codes over the ring ${\mathscr{B}}_{j,k}$

Autor: Serap Sahinkaya, Joe Gildea, Steven T. Dougherty, Adrian Korban
Rok vydání: 2021
Předmět:
Zdroj: Cryptography and Communications. 13:601-616
ISSN: 1936-2455
1936-2447
DOI: 10.1007/s12095-021-00487-x
Popis: In this work, we study a new family of rings, ${\mathscr{B}}_{j,k}$ B j , k , whose base field is the finite field ${\mathbb {F}}_{p^{r}}$ F p r . 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 ${\mathscr{B}}_{j,k}$ B j , k to a code over ${\mathscr{B}}_{l,m}$ B l , 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 $G^{2^{j+k}}$ G 2 j + 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.
Databáze: OpenAIRE
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