  The design of a four-dimensional toric code is explored with the goal of
finding a lattice capable of implementing a logical $\mathsf{CCCZ}$ gate
transversally. The established lattice is the octaplex tessellation, which is a
regular tessellation of four-dimensional Euclidean space whose underlying
4-cell is the octaplex, or hyper-diamond. This differs from the conventional 4D
toric code lattice, based on the hypercubic tessellation, which is symmetric
with respect to logical $X$ and $Z$ and only allows for the implementation of a
transversal Clifford gate. This work further develops the established
connection between topological dimension and transversal gates in the Clifford
hierarchy, generalizing the known designs for the implementation of transversal
$\mathsf{CZ}$ and $\mathsf{CCZ}$ in two and three dimensions, respectively.
