Bulletin of the
Korean Mathematical Society BKMS

In this paper, we introduce $SR$-additive codes as a generalization of the classes of $\mathbb{Z}_{p^r}\mathbb{Z}_{p^s}$ and $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-additive codes, where $S$ is an $R$-algebra and an $SR$-additive code is an $R$-submodule of $S^{\alpha}\times R^{\beta}$. In particular, the definitions of bilinear forms, weight functions and Gray maps on the classes of $\mathbb{Z}_{p^r}\mathbb{Z}_{p^s}$ and $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-additive codes are generalized to $SR$-additive codes. Also the singleton bound for $SR$-additive codes and some results on one weight $SR$-additive codes are given. Among other important results, we obtain the structure of $SR$-additive cyclic codes. As some results of the theory, the structure of cyclic $\mathbb{Z}_{2}\mathbb{Z}_{4}$, $\mathbb{Z}_{p^r}\mathbb{Z}_{p^s}$, $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$, $(\mathbb{Z}_{2})(\mathbb{Z}_{2} + u\mathbb{Z}_{2} + u^{2}\mathbb{Z}_{2})$, $(\mathbb{Z}_{2} + u\mathbb{Z}_{2} )(\mathbb{Z}_{2} + u\mathbb{Z}_{2} + u^{2}\mathbb{Z}_{2})$, $(\mathbb{Z}_{2})(\mathbb{Z}_{2} + u\mathbb{Z}_{2} + v\mathbb{Z}_{2})$ and $(\mathbb{Z}_{2} + u\mathbb{Z}_{2} )(\mathbb{Z}_{2} + u\mathbb{Z}_{2} + v\mathbb{Z}_{2})$-additive codes are presented.

Keywords: additive code, chain ring, Galois ring

MSC numbers: 94B15