Rational points of lattice ideals on a toric variety and toric codes

We show that the number of rational points of a subgroup inside a toric variety over a finite field defined by a homoge-neous 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 appli-cation, we compute the main parameters of generalized toric codes on subgroups of the torus of Hirzebruch surfaces, gen-eralizing the existing literature. (c) 2023 Elsevier Inc. All rights reserved.