I'm going to try this instead of Halton sequences in my current project.

One detail of the implementation that may be worth improving:

        z[i] = (seed + alpha*(i+1)) %1

When i gets large, say 2^30, you're losing 30 bits of precision in the %1, so (with IEEE doubles) it can only take one of 2^23 values.

The usual implementation of Halton sequences get slower with large values of i, but doesn't suffer from quantization.