Vanishing ideals of parameterized subgroups in a toric variety

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