  We show that the number of rational points of a subgroup inside a toric
variety over a finite field defined by a homogeneous lattice ideal can be
computed via Smith normal form of the matrix whose columns constitute a basis
of the lattice. This generalizes and yields a concise toric geometric proof of
the same fact proven purely algebraically by Lopez and Villarreal for the case
of a projective space and a standard homogeneous lattice ideal of dimension
one. We also prove a Nullstellensatz type theorem over a finite field
establishing a one to one correspondence between subgroups of the dense split
torus and certain homogeneous lattice ideals. As application, we compute the
main parameters of generalized toric codes on subgroups of the torus of
Hirzebruch surfaces, generalizing the existing literature.
