Vanishing Ideals of Parameterized Subgroups in a toric variety

Let \(\K\) be a finite field and \(X\) be a complete simplicial toric variety over \(\K\). 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 \verb|Macaulay2| codes for our algorithms.