  This paper is devoted to the study of lattices generated by finite Abelian
groups. Special species of such lattices arise in the exploration of elliptic
curves over finite fields. In case the generating group is cyclic, they are
also known as the Barnes lattices. It is shown that for every finite Abelian
group with the exception of the cyclic group of order four these lattices have
a basis of minimal vectors. Another result provides an improvement of a recent
upper bound by Min Sha for the covering radius in the case of the Barnes
lattices. Also discussed are properties of the automorphism groups of these
lattices.
