Additive conjucyclic codes over a class of Galois rings

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_q^2 ) 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 2 n over ℤ_2^r . This correspondence helps to obtain additive conjucyclic codes over GR(2^r,2) via known linear cyclic codes over ℤ_2^r . 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_2^r contained in additive conjucyclic codes over GR(2^r,2) . Last but not least, we explicitly derive the generator matrices for these codes.