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.