论文标题
一类很少的糖重量$ \ mathbb {z} _2 [u] $ - 使用Simplicial复合物和最小代码通过灰色地图进行线性代码
A class of few-Lee weight $\mathbb{Z}_2[u]$-linear codes using simplicial complexes and minimal codes via Gray map
论文作者
论文摘要
最近，一些混合字母环参与使用合适的定义集或下组构建具有最佳或最小灰色图像的少量质量添加代码。受这些作品的启发，我们选择了混合字母环$ \ Mathbb {z} _2 \ Mathbb {z} _2 [u] $来构造一类特殊的线性代码$ c_l $ c_l $ c_l $ c_l $ c_l $ c_l $ \ mathbb {z} _2 _2 _2 _2 [u^2 [u] $ a用$ u^2 = 0 $ a用$ u^u^2 = 0 $ a用简化的复合物生成的元素由单个maxim axcimim emplecim axcimim emaxim emaxim e元素。我们表明，$ C_L $通过确定$ C_L $的LEE权重分布，具有很少的糖分权重。从理论上讲，这表明即使在混合字母环的情况下，我们也可以使用简单的复合物来实现很少的重量代码。我们表明，$ C_L $的灰色图像是自动的，并且通过灰色地图，我们拥有$ \ Mathbb {z} _2 $的无限族家族，可用于秘密共享方案。
Recently some mixed alphabet rings are involved in constructing few-Lee weight additive codes with optimal or minimal Gray images using suitable defining sets or down-sets. Inspired by these works, we choose the mixed alphabet ring $\mathbb{Z}_2\mathbb{Z}_2[u]$ to construct a special class of linear code $C_L$ over $\mathbb{Z}_2[u]$ with $u^2=0$ by employing simplicial complexes generated by a single maximal element. We show that $C_L$ has few-Lee weights by determining the Lee weight distribution of $C_L$. Theoretically, this shows that we may employ simplicial complexes to obatin few-weight codes even in the case of mixed alphabet rings. We show that the Gray image of $C_L$ is self-orthogonal and we have an infinite family of minimal codes over $\mathbb{Z}_2$ via Gray map, which can be used to secret sharing schemes.
