  Quantum error correction (QEC) plays a critical role in preventing
information loss in quantum systems and provides a framework for reliable
quantum computation. Identifying quantum codes with nice code parameters for
physically motivated noise models remains an interesting challenge. Going
beyond qubit codes, here we propose a class of qudit error correcting codes
tailored to protect against amplitude-damping noise. Specifically, we construct
a class of four-qudit codes that satisfies the error correction conditions for
all single-qudit and a few two-qudit damping errors up to the leading order in
the damping parameter $\gamma$. We devise a protocol to extract syndromes that
identify this set of errors unambiguously, leading to a noise-adapted recovery
scheme that achieves a fidelity loss of $\cO(\gamma^{2})$. For the $d=2$ case,
our QEC scheme is identical to the known example of the $4$-qubit code and the
associated syndrome-based recovery.
  We also assess the performance of our class of codes using the Petz recovery
map and note some interesting deviations from the qubit case.
