G -codes, self-dualG -codes and reversibleG -codes over the ring B_(j,k)

摘要

In this work, we study a new family of rings, B-j,B-k,B- whose base field is the finite field F-pr. 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 Gcodes, self-dual G-codes, and reversible G-codes over this family. In particular, we show that the projection of a G-code over B-j,B-k to a code over B-l,B-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 G2(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.