# Do low-discrepancy sequences work in discrete spaces?

Low-discrepancy sequences in a real space ($[0,1]^n$) seem like a really excellent tool for evenly sampling a sample space. As far as I can tell, they generalise well to any real space, if you use an appropriate map (eg. $[0,1]\to[a,b]$ linear map).

Do such sequences generalise to discrete spaces? eg. if I have a space that has only two elements in each dimension (eg. boolean switches), can I just map $[0,0.5]\to 0;\ (0.5,1]\to 1$? What about for dimensions with more elements (eg. a 4-state switch?). And for spaces with different number of states in each dimension?

My intuition says that this might work ok, especially if the sub sequence is longish, but that it might work better for some sequences than for others, depending on the number of states (eg. a Halton sequence might have odd interactions with a dimension with a prime number of states, or a Sobol' sequence might only work for dimensions with $2^n$ elements). But I have done no testing.

If this doesn't work, why not?

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If you have a finite number of spaces, you will be better off with an explicit enumeration of possible spaces with a balanced incomplete block design built upon them. In the end, the properties of the low discrepancy sequences are asymptotic, with desirable properties achieved with the lengths of the order $N\sim 6^s$ where $s$ is the dimension of your space. If the number of possible combinations is less than that, you can just take all possible combination and achieve a balanced design with that.