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 ${\mathbb{R}}^2$.

Contents IJAM: Volume 32, No. 2 (2019)

Contents IJAM: Volume 31, No. 1 (2018)

Contents IJAM: Volume 31, No. 4 (2018)