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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.
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참고문헌 (23)
참고문헌 신청
Taher Abualrub / 2007 / Cyclic codes over the rings Z 2 + uZ 2 and Z 2 + uZ 2 + u 2 Z 2 / Designs, Codes and Cryptography 42(3):273~287
Taher Abualrub / 2014 / $\BBZ_{2}\BBZ_{4}$ -Additive Cyclic Codes / IEEE Transactions on Information Theory 60(3):1508~1514
Ismail Aydogdu / 2014 / On ℤ 2 ℤ 2 [ u ]-additive codes / International Journal of Computer Mathematics 92(9):1806~1814
I. Aydogdu / 2013 / The structure of Z2Z2s -additive codes: bounds on the minimum distance / Appl. Math. Inf. Sci 7(6):2271~2278
Ismail Aydogdu / 2014 / On ℤ p r ℤ p s -additive codes / Linear and Multilinear Algebra 63(10):2089~2102
Taher Abualrub / 2014 / $\BBZ_{2}\BBZ_{4}$ -Additive Cyclic Codes / IEEE Transactions on Information Theory 60(3):1508~1514
Ismail Aydogdu / 2014 / On ℤ 2 ℤ 2 [ u ]-additive codes / International Journal of Computer Mathematics 92(9):1806~1814
I. Aydogdu / 2013 / The structure of Z2Z2s -additive codes: bounds on the minimum distance / Appl. Math. Inf. Sci 7(6):2271~2278
Ismail Aydogdu / 2014 / On ℤ p r ℤ p s -additive codes / Linear and Multilinear Algebra 63(10):2089~2102
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