I am studying Monte-Carlo simulations using quasi random numbers and encounter the following problem: I am given a set of 1D quasi-random numbers $(X_i)$ over $[0,1)$, and would like to generate another set of 1D quasi-random numbers $(Y_i)$ over $[0,1)$, that jointly form a set $(X_i,Y_i)$ of quasi-random numbers over $[0,1)^2$. In other words, I would like Y to be "independent" of X. This is trivial with pseudo-random numbers, but for quasi numbers this is usually not the case. In fact, Y may be exactly the same as X if one generates Y using the same algorithm as X.
More generally, how do I sequentially generate sets of independent quasi-numbers? This is necessary for simulation a Brownian motion, but so far I have not seen how this can be done. Even if one could generate 2 set of independent quasi numbers, I don't see how one could generate indefinitely many sets. Any ideas?
Thank you for your inputs.