Generalized Additive Codes Over Galois Rings

Additive codes were first defined by Delsarte in1973. An additive code is a subgroup of the underlying Abelian group of the form Zα2×Zβ4, where α+2β=n, which is also called Z2Z4-additive code. Every codeword of Z2Z4-additive codes is a vector of length α+β, and of which the former α elements come from Z2and the later β elements come from Z4. The code obtained when a=0is called a quaternary linear code. The code obtained when β=0is called a binary linear code.In this paper, we will generalize Z2Z4-additive codes to Galois rings. We call the codes as generalized additive codes. The generalized additive code is a submodule of the Z4[ξ]-module Z2[ξ]α×Z4[ξ]β. Every codeword of generalized additive codes is a vector of length α+β, and of which the former a elements come from Z2[ξ] and the later β elements come from Z4[ξ].The generalized additive codes and their dual codes over Galois ring are studied in this paper. The fundamental parameters of these codes are obtained and their standard forms for generator matrices are also given. We also study the Singleton bound of Lee metric of generalized additive codes.