Toric Codes and Lattice Ideals

Let X be a complete simplicial toric variety over a finite field Fq with homogeneous coordinate ring S = Fq[x1, . . . , xr] and split torus TX ∼= (Fq). We prove that vanishing ideal of a subset Y of the torus TX is a lattice ideal if and only if Y is a subgroup. We show that these subgroups are exactly those subsets that are parameterized by Laurents monomials. We give an algorithm for determining this parametrization if the subgroup is the zero locus of a lattice ideal in the torus. We also show that vanishing ideals of subgroups of TX are radical homogeneous lattice ideals of dimension r − n. We identify the lattice corresponding to a degenerate torus in X and completely characterize when its lattice ideal is a complete intersection. We compute dimension and length of some generalized toric codes defined on these degenerate tori.