Vanishing ideals of parameterized subgroups in a toric variety

Let K be a finite field and X be a complete simplicial toric variety over K with split torus T-X congruent to (K*)(n). We give an algorithm relying on elimination theory for finding generators of the vanishing ideal of a subgroup Y-Q parameterized by a matrix Q which can be used to study algebraic geometric codes arising from Y-Q. We give a method to compute the lattice L whose ideal I-L is exactly I(Y-Q) under a mild condition. As applications, we give precise descriptions for the lattices corresponding to some special subgroups. We also prove a Nullstellensatz type theorem valid over finite fields, and share Macaulay2* codes for our algorithms.