  Designing molecules with desirable physiochemical properties and
functionalities is a long-standing challenge in chemistry, material science,
and drug discovery. Recently, machine learning-based generative models have
emerged as promising approaches for \emph{de novo} molecule design. However,
further refinement of methodology is highly desired as most existing methods
lack unified modeling of 2D topology and 3D geometry information and fail to
effectively learn the structure-property relationship for molecule design. Here
we present MolCode, a roto-translation equivariant generative framework for
\underline{Mol}ecular graph-structure \underline{Co-de}sign. In MolCode, 3D
geometric information empowers the molecular 2D graph generation, which in turn
helps guide the prediction of molecular 3D structure. Extensive experimental
results show that MolCode outperforms previous methods on a series of
challenging tasks including \emph{de novo} molecule design, targeted molecule
discovery, and structure-based drug design. Particularly, MolCode not only
consistently generates valid (99.95$\%$ Validity) and diverse (98.75$\%$
Uniqueness) molecular graphs/structures with desirable properties, but also
generate drug-like molecules with high affinity to target proteins (61.8$\%$
high-affinity ratio), which demonstrates MolCode's potential applications in
material design and drug discovery. Our extensive investigation reveals that
the 2D topology and 3D geometry contain intrinsically complementary information
in molecule design, and provide new insights into machine learning-based
molecule representation and generation.
