  Quantum codes typically rely on large numbers of degrees of freedom to
achieve low error rates. However each additional degree of freedom introduces a
new set of error mechanisms. Hence minimizing the degrees of freedom that a
quantum code utilizes is helpful. One quantum error correction solution is to
encode quantum information into one or more bosonic modes. We revisit
rotation-invariant bosonic codes, which are supported on Fock states that are
gapped by an integer $g$ apart, and the gap $g$ imparts number shift resilience
to these codes. Intuitively, since phase operators and number shift operators
do not commute, one expects a trade-off between resilience to number-shift and
rotation errors. Here, we obtain results pertaining to the non-existence of
approximate quantum error correcting $g$-gapped single-mode bosonic codes with
respect to Gaussian dephasing errors. We show that by using arbitrarily many
modes, $g$-gapped multi-mode codes can yield good approximate quantum error
correction codes for any finite magnitude of Gaussian dephasing and amplitude
damping errors.
