  Quantum error correction (QEC) plays an essential role in fault-tolerantly
realizing quantum algorithms of practical interest. Among different approaches
to QEC, encoding logical quantum information in harmonic oscillator modes has
been shown to be promising and hardware efficient. In this work, we study
multimode Gottesman-Kitaev-Preskill (GKP) codes, encoding a qubit in many
oscillators, through a lattice perspective. In particular, we implement a
closest point decoding strategy for correcting random Gaussian shift errors.
For decoding a generic multimode GKP code, we first identify its corresponding
lattice followed by finding the closest lattice point in its symplectic dual
lattice to a candidate shift error compatible with the error syndrome. We use
this method to characterize the error correction capabilities of several known
multimode GKP codes, including their code distances and fidelities. We also
perform numerical optimization of multimode GKP codes up to ten modes and find
three instances (with three, seven and nine modes) with better code distances
and fidelities compared to the known GKP codes with the same number of modes.
While exact closest point decoding incurs exponential time cost in the number
of modes for general unstructured GKP codes, we give several examples of
structured GKP codes (i.e., of the repetition-rectangular GKP code types) where
the closest point decoding can be performed exactly in linear time. For the
surface-GKP code, we show that the closest point decoding can be performed
exactly in polynomial time with the help of a minimum-weight-perfect-matching
algorithm (MWPM). We show that this MWPM closest point decoder improves both
the fidelity and the noise threshold of the surface-GKP code to 0.602 compared
to the previously studied MWPM decoder assisted by log-likelihood analog
information which yields a noise threshold of 0.599.
