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We introduce a method for constructing a rich class of designs that are suitable for use in computer experiments. The designs include Latin hypercube designs and two-level fractional factorial designs as special cases and fill the vast vacuum between these two familiar classes of designs. The basic construction method is simple, building a series of larger designs based on a given small design. If the base design is orthogonal, the resulting designs are orthogonal; likewise, if the base design is nearly orthogonal, the resulting designs are nearly orthogonal. We present two generalizations of our basic construction method. The first generalization improves the projection properties of the basic method; the second generalization gives rise to designs that have smaller correlations. Sample constructions are presented and properties of these designs are discussed.