# Are non-square latin hypercubes viable?

At https://github.com/OpenMDAO/OpenMDAO-Framework/issues/599 it is stated that non-square Latin Hypercube experimental design is not well defined (I assume that for higher dimensions that means hypercube must have the same length in every dimension).

I'm wondering why that is. Would it not be possible to, for example, have a LHC scheme where each row in each dimension had one or more samples, and the sample space was relatively evenly sampled, with no correlation between variables? I guess this is actually making a hypercube out of a hyperrectangle, but is there any reason that wouldn't work?

For example, I have a bunch of parameters that I want to change in a physical model. Some have 4 states, some only two. Could I make a hypercube where the Parameters with only two states just have those states duplicated (so that there are two model runs with each of those two states)? how about doing the same with 4-, 3- and 2-state parameters? Could I make an LHC scheme where each dimension has length equal to the smallest common multiple of the states count for each parameter (ie. 12)?

Remember: The basic idea of LHS sampling is to sample with equi-probabilistically. i.e. split a pdf into a equiprobable areas and take a sample from each one. For hyper cubes or rectangles this is quite easy. Just break your $$U[a,b]$$ uniform distribution into $$N$$ intervals: $$(b-a)/N$$.