  We give a general construction of entire functions in $d$ complex variables
that vanish on a lattice of the form $L = A (Z + i Z )^d$ for an invertible
complex-valued matrix. As an application we exhibit a class of lattices of
density >1 that fail to be a sampling set for the Bargmann-Fock space in $C
^2$. By using an equivalent real-variable formulation, we show that these
lattices fail to generate a Gabor frame.
