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Some results on \begin{document}$ \mathbb{Z}_p\mathbb{Z}_p[v] $\end{document}-additive cyclic codes
Advances in Mathematics of Communications ( IF 0.9 ) Pub Date : 2019-09-04 , DOI: 10.3934/amc.2020029
Lingyu Diao , , Jian Gao , Jiyong Lu , , ,

$ \mathbb{Z}_p\mathbb{Z}_p[v] $-Additive cyclic codes of length $ (\alpha,\beta) $ can be viewed as $ R[x] $-submodules of $ \mathbb{Z}_p[x]/(x^\alpha-1)\times R[x]/(x^\beta-1) $, where $ R = \mathbb{Z}_p+v\mathbb{Z}_p $ with $ v^2 = v $. In this paper, we determine the generator polynomials and the minimal generating sets of this family of codes as $ R[x] $-submodules of $ \mathbb{Z}_p[x]/(x^\alpha-1)\times R[x]/(x^\beta-1) $. We also determine the generator polynomials of the dual codes of $ \mathbb{Z}_p\mathbb{Z}_p[v] $-additive cyclic codes. Some optimal $ \mathbb{Z}_p\mathbb{Z}_p[v] $-linear codes and MDSS codes are obtained from $ \mathbb{Z}_p\mathbb{Z}_p[v] $-additive cyclic codes. Moreover, we also get some quantum codes from $ \mathbb{Z}_p\mathbb{Z}_p[v] $-additive cyclic codes.

中文翻译:

一些结果 \ begin {document} $ \ mathbb {Z} _p \ mathbb {Z} _p [v] $ \ end {document}附加循环码

$ \ mathbb {Z} _p \ mathbb {Z} _p [v] $-长度为$(\ alpha,\ beta)$的加性循环码可以看作$ \ mathbb {Z的$ R [x] $-子模块} _p [x] /(x ^ \ alpha-1)\ R R [x] /(x ^ \ beta-1)$,其中$ R = \ mathbb {Z} _p + v \ mathbb {Z} _p $其中$ v ^ 2 = v $。在本文中,我们将生成器多项式和该代码家族的最小生成集确定为$ \ mathbb {Z} _p [x] /(x ^ \ alpha-1)\ times的$ R [x] $-子模块R [x] /(x ^ \ beta-1)$。我们还确定了$ \ mathbb {Z} _p \ mathbb {Z} _p [v] $可加循环码对偶代码的生成多项式。从$ \ mathbb {Z} _p \ mathbb {Z} _p [v] $加性循环码中获得一些最优的$ \ mathbb {Z} _p \ mathbb {Z} _p [v] $线性码和MDSS码。此外,我们还从$ \ mathbb {Z} _p \ mathbb {Z} _p [v] $可加循环码中获得一些量子码。
更新日期:2019-09-04
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