  Latin squares and hypercubes are combinatorial designs with several
applications in statistics, cryptography and coding theory. In this paper, we
generalize a construction of Latin squares based on bipermutive cellular
automata (CA) to the case of Latin hypercubes of dimension $k>2$. In
particular, we prove that linear bipermutive CA (LBCA) yielding Latin
hypercubes of dimension $k>2$ are defined by sequences of invertible Toeplitz
matrices with partially overlapping coefficients, which can be described by a
specific kind of regular de Bruijn graph induced by the support of the
determinant function. Further, we derive the number of $k$-dimensional Latin
hypercubes generated by LBCA by counting the number of paths of length $k-3$ on
this de Bruijn graph.
