Cover the space evenly instead of drawing from it evenly.
What separates it from Random is not what it looks at —
both look at nothing — but what it promises. Random is uniform in
expectation; this is uniform by construction, for every prefix: of the
first base² trials exactly one lands in each cell of a base² grid, and no
arrangement of the indices makes it otherwise. For a study handed out of a
folder that is the difference between a collision being unlikely and there
being no way to arrange one.
Knob d is read in base the d-th prime, which is why the promise thins out
once there are many knobs: the high primes need a long prefix before they
look like anything. Sobol has no such seam, at the price of
a table. Its point is a function of the seed and the index.
The seed, which here permutes the digits rather than drawing them: with
no scramble a Halton sequence is one fixed sequence, and two studies of
the same space would search it in exactly the same order.