In this paper we present Gröbner bases for lattices. Gröbner bases for binary linear codes were introduced by Borges et al. . We extend their work to non-binary group block codes. Given a lattice Λ and its associated label code L, which is a group code, we define an ideal for L. A Gröbner basis is assigned to Λ as the Gröbner basis of its label code L. Using this Gröbner basis an algebraic decoding algorithm is introduced.