$SR$-additive codes
Bull. Korean Math. Soc.
Published online August 6, 2019
Saadoun Mahmoudi and Karim Samei
Bu Ali Sina
Abstract : ‎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
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