Additive conjucyclic codes over a class of Galois rings

As a tool towards quantum error correction, additive conjucyclic codes have gained great attention. But, their algebraic structure is completely unknown over finite fields (except F-q2) as well as rings. In this article, we investigate the structure of additive conjucyclic codes over Galois rings GR(2(r), 2), where r >= 2 is an integer. We develop a one-to-one correspondence between the family of additive conjucyclic codes of length n over GR(2(r), 2) and the family of linear cyclic codes of length 2n over Z(2r). This correspondence helps to obtain additive conjucyclic codes over GR(2(r), 2) via known linear cyclic codes over Z(2r). We prove that the trace dual C Tr of an additive conjucyclic code C is also an additive conjucyclic code. Moreover, we derive a necessary and sufficient condition of additive conjucyclic codes to be self-dual. We further propose a technique for constructing linear cyclic codes over Z(2r) contained in additive conjucyclic codes over GR(2(r), 2). Last but not least, we explicitly derive the generator matrices for these codes.