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⁎)n. We prove that submonoids of TX are exactly those subsets that are parameterized by Laurents monomials. We give an algorithm for determining this parametrization if the submonoid is the zero locus of a lattice ideal in the torus. We also show that vanishing ideals of submonoids 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.