G-codes, self-dual G-codes and reversible G-codes over the ring ${\mathscr{B}}_{j,k}$
Tóm tắt
In this work, we study a new family of rings,
${\mathscr{B}}_{j,k}$
, whose base field is the finite field
${\mathbb {F}}_{p^{r}}$
. 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
${\mathscr{B}}_{j,k}$
to a code over
${\mathscr{B}}_{l,m}$
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
$G^{2^{j+k}}$
-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.
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