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

In this work, we show that an n-dimensional sublattice Λ′=mΛ of an n-dimensional lattice Λ induces a G=Zmn tessellation in the flat torus Tβ′=Rn/Λ′, where the group G is isomorphic to the lattice partition Λ/Λ′. As a consequence, we obtain, via this technique, toric codes of parameters [[2m2,2,m]], [[3m3,3,m]] and [[6m4,6,m2]] from the lattices Z2, Z3 and Z4, respectively. In particular, for n=2, if Λ1 is either the lattice Z2 or a hexagonal lattice, through lattice partition, we obtain two equivalent ways to cover the fundamental cell P0′ of each hexagonal sublattice Λ′ of hexagonal lattices Λ, using either the fundamental cell P0 or the Voronoi cell V0. These partitions allow us to present new classes of toric codes with parameters [[3m2,2,m]] and color codes with parameters [[18m2,4,4m]] in the flat torus from families of hexagonal lattices in R2.

Projective toric codes

Any integral convex polytope [Formula: see text] in [Formula: see text] provides an [Formula: see text]-dimensional toric variety [Formula: see text] and an ample divisor [Formula: see text] on this variety. This paper gives an explicit construction of the algebraic geometric error-correcting code on [Formula: see text], obtained by evaluating global section of the line bundle corresponding to [Formula: see text] on every rational point of [Formula: see text]. This work presents an extension of toric codes analogous to the one of Reed–Muller codes into projective ones, by evaluating on the whole variety instead of considering only points with nonzero coordinates. The dimension of the code is given in terms of the number of integral points in the polytope [Formula: see text] and an algorithmic technique to get a lower bound on the minimum distance is described.