We study the properties of quantum stabilizer codes that embed a finite-dim
ensional protected code space in an infinite-dimensional Hilbert space. The
stabilizer group of such a code is associated with a symplectically integr
al lattice in the phase space of 2N canonical variables. From the existence
of symplectically integral lattices with suitable properties, we infer a l
ocker bound on the quantum capacity of the Gaussian quantum channel that ma
tches the one-shot coherent information optimized over Gaussian input state
s.