Latin hypercube designs (LHDs), widely used for computer experiments, are a very large class of designs with desirable properties. Recently, a number of methods have been proposed to construct orthogonal LHDs. In this paper, we in- troduce an approach to constructing 2 r -order orthogonal designs. The methods are simple and easy to implement. Using orthogonal designs, we propose some methods for constructing orthogonal and nearly orthogonal LHDs so that the elementwise square of each column and the elementwise product of any two distinct columns are orthogonal to all columns. Further, the resulting nearly orthogonal LHDs with 2 r+1 +2 runs and 2 r factors have the minimum correlation between any two distinct columns.