  The Gottesman-Kitaev-Preskill (GKP) code offers the possibility to encode
higher-dimensional qudits into individual bosonic modes with, for instance,
photonic excitations. Since photons enable the reliable transmission of quantum
information over long distances and since GKP states subject to photon loss can
be recovered to some extent, the GKP code has found recent applications in
theoretical investigations of quantum communication protocols. While previous
studies have primarily focused on GKP qubits, the possible practical benefits
of higher-dimensional GKP qudits are hitherto widely unexplored. In this paper,
we carry out performance analyses for three quantum repeater protocols based on
GKP qudits including concatenations with a multi-qudit quantum polynomial code.
We find that the potential data transmission gains for qudits are often
hampered by their decreased GKP error-correcting capabilities. However, we also
identify parameter regimes in which having access to an increased number of
quantum levels per mode can enhance the theoretically achievable secret-key
rate of the quantum repeater. Some of our protocols share the attractive
feature that local processing and complete error syndrome identification are
realizable without online squeezing. Provided a supply of suitable multi-mode
GKP states is available, this can be realized with a minimal set of passive
linear optical operations, even when the logical qudits are composed of many
physical qudits.
