  The Gottesman-Kitaev-Preskill (GKP) code encodes a qubit into a bosonic mode
using periodic wavefunctions. This periodicity makes the GKP code a natural
setting for the Zak transform, which is tailor-made to provide a simple
description for periodic functions. We review the Zak transform and its
connection to a Zak basis of states in Hilbert space, decompose the shift
operators that underpin the stabilizers and the correctable errors, and we find
that Zak transforms of the position wavefunction appear naturally in GKP error
correction. We construct a new bosonic subsystem decomposition (SSD) -- the
modular variable SSD -- by dividing a mode's Hilbert space, expressed in the
Zak basis, into that of a virtual qubit and a virtual gauge mode. Tracing over
the gauge mode gives a logical-qubit state, and preceding the trace with a
particular logical-gauge interaction gives a different logical state -- that
associated to GKP error correction.
