In this work, we show that an n-dimensional sublattice ${\Lambda }^{\prime }=m\Lambda$ of an n-dimensional lattice $\Lambda$ induces a $G={\mathbb{Z}}_m^n$ tessellation in the flat torus $T_{{\beta }^{\prime }}={\mathbb{R}}^n/{\Lambda }^{\prime }$, where the group G is isomorphic to the lattice partition $\Lambda /{\Lambda }^{\prime }$. As a consequence, we obtain, via this technique, toric codes of parameters $\left[[2m^2,2,m]\right]$, $\left[[3m^3,3,m]\right]$ and $\left[[6m^4,6,m^2]\right]$ from the lattices ${\mathbb{Z}}^2$, ${\mathbb{Z}}^3$ and ${\mathbb{Z}}^4$, respectively. In particular, for $n=2$, if ${\Lambda }_1$ is either the lattice ${\mathbb{Z}}^2$ or a hexagonal lattice, through lattice partition, we obtain two equivalent ways to cover the fundamental cell ${\mathcal{P}}_0^{\prime }$ of each hexagonal sublattice ${\Lambda }^{\prime }$ of hexagonal lattices $\Lambda$, using either the fundamental cell ${\mathcal{P}}_0$ or the Voronoi cell ${\mathcal{V}}_0$. These partitions allow us to present new classes of toric codes with parameters $\left[[3m^2,2,m]\right]$ and color codes with parameters $\left[[18m^2,4,4m]\right]$ in the flat torus from families of hexagonal lattices in ${\mathbb{R}}^2$.

中文翻译：

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n维平面圆环上晶格划分的拓扑量子码

在这项工作中，我们证明了一个n维亚晶格${\Lambda }^{\prime }=米\Lambda$一个的Ñ维晶格$\Lambda$ 诱导一个 $G={\mathbb{Z}}_{米}^n$ 平面圆环中的镶嵌 ${吨}_{{\beta }^{\prime }}={\mathbb{电阻}}^n/{\Lambda }^{\prime }$，其中群G与格子划分同构$\Lambda /{\Lambda }^{\prime }$. 因此，我们通过这种技术获得了参数的复曲面代码$\left[[2{米}^2,2,米]\right]$, $\left[[3{米}^3,3,米]\right]$ 和 $\left[[6{米}^4,6,{米}^2]\right]$ 从格子 ${\mathbb{Z}}^2$, ${\mathbb{Z}}^3$ 和 ${\mathbb{Z}}^4$， 分别。特别是，对于$n=2$， 如果 ${\Lambda }_1$ 是格子 ${\mathbb{Z}}^2$ 或六边形点阵，通过点阵划分，我们得到两种等价的方式来覆盖基本单元格 ${\mathcal{磷}}_0^{\prime }$ 每个六边形亚晶格 ${\Lambda }^{\prime }$ 六边形格子 $\Lambda$, 使用基本单元格 ${\mathcal{磷}}_0$ 或 Voronoi 细胞 ${\mathcal{伏}}_0$. 这些分区允许我们用参数呈现新的复曲面代码类别$\left[[3{米}^2,2,米]\right]$ 和带参数的颜色代码 $\left[[18{米}^2,4,4米]\right]$ 在来自六边形格子家族的扁平圆环中 ${\mathbb{电阻}}^2$.