Topological Quantum Codes from Lattices Partition on the n-Dimensional Flat Tori

In this work, we show that an n-dimensional sublattice [Formula: see text] of an n-dimensional lattice [Formula: see text] induces a [Formula: see text] tessellation in the flat torus [Formula: see text] , where the group G is isomorphic to the lattice partition [Formula: see text]. As a consequence, we obtain, via this technique, toric codes of parameters [Formula: see text] , [Formula: see text] and [Formula: see text] from the lattices [Formula: see text] , [Formula: see text] and [Formula: see text] , respectively. In particular, for [Formula: see text] , if [Formula: see text] is either the lattice [Formula: see text] or a hexagonal lattice, through lattice partition, we obtain two equivalent ways to cover the fundamental cell [Formula: see text] of each hexagonal sublattice [Formula: see text] of hexagonal lattices [Formula: see text] , using either the fundamental cell [Formula: see text] or the Voronoi cell [Formula: see text]. These partitions allow us to present new classes of toric codes with parameters [Formula: see text] and color codes with parameters [Formula: see text] in the flat torus from families of hexagonal lattices in [Formula: see text].