Some results on linear codes over the ring $$\mathbb {Z}_4+u\mathbb {Z}_4+v\mathbb {Z}_4+uv\mathbb {Z}_4$$ |
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Authors: | Ping Li Xuemei Guo Shixin Zhu Xiaoshan Kai |
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Institution: | 1.School of Mathematics,Hefei University of Technology,Hefei,People’s Republic of China;2.National Mobile Communications Research Laboratory,Southeast University,Nanjing,People’s Republic of China |
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Abstract: | In this paper, we mainly study the theory of linear codes over the ring \(R =\mathbb {Z}_4+u\mathbb {Z}_4+v\mathbb {Z}_4+uv\mathbb {Z}_4\). By using the Chinese Remainder Theorem, we prove that R is isomorphic to a direct sum of four rings. We define a Gray map \(\Phi \) from \(R^{n}\) to \(\mathbb {Z}_4^{4n}\), which is a distance preserving map. The Gray image of a cyclic code over R is a linear code over \(\mathbb {Z}_4\). We also discuss some properties of MDS codes over R. Furthermore, we study the MacWilliams identities of linear codes over R and give the generator polynomials of cyclic codes over R. |
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