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 G^s-codes for some s.

在这项工作中，我们研究了一个新的环族\（{\ mathscr {B}} _ {j，k} \），其基本字段是有限字段\（{\ mathbb {F}} _ {p ^ {r}} \）。我们研究了该环族的结构，并表明该族的每个成员都是可交换的Frobenius环。我们为新的戒指族定义了一个灰色图，研究该族的G代码，自对偶G代码和可逆G代码。特别是，我们证明了G代码在\（{\ mathscr {B}} _ {j，k} \）上的投影到代码在\（{\ mathscr {B}} _ {l，m}上的投影\）也是G代码，并且是自对偶G的Gray映射下的图像基字段的特征为2时，-code也是自对偶G - code。此外，我们证明了Gray映射下可逆G- code的图像也是可逆\（G ^ {2 ^ { j + k}} \） -代码。这些代码的格雷图像显示具有丰富的自同构群，该自同构群是由环和群的代数结构引起的。最后，我们表明，准摹代码，这是的图像摹灰色地图下-codes，也摹^小号一些-codes小号。