We propose an approach for constructing a new type of design, called a sliced orthogonal array-based Latin hypercube design. This approach exploits a slicing structure of orthogonal arrays with strength two and makes use of sliced random permutations. Such a design achieves one- and two-dimensional uniformity and can be divided into smaller Latin hypercube designs with one-dimensional uniformity. Sampling properties of the proposed designs are derived. Examples are given for illustrating the construction method and corroborating the derived theoretical results. Potential applications of the constructed designs include uncertainty quantification of computer models, computer models with qualitative and quantitative factors, cross-validation and efficient allocation of computing resources. Supplementary materials for this article are available online.