Toric codes and lattice ideals

Let X be a complete simplicial toric variety over a finite field F-q with homogeneous coordinate ring S = F-q [x(1),., x(r)] and split torus T-X congruent to (F*(q))(n). We prove that submonoids of T-X 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 T-X 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. (C) 2018 Elsevier Inc. All rights reserved.