Another property of random number sequences is that there are no long
autocorrelations. A 64k snippet, or any other snippet for that matter will
eventually have a 100% overall autocorrelation. You may rethink your reasons
for needing random numbers and decide that pseudorandom numbers will do.
They are easy to generate algorithmically as you already have discovered.
The most economical solution that I have seen so far is the diode scheme.
Once that is generated then sampling it with an analog to digital converter
to the desired resolution is all that is needed. This should result in a
totally noncorrelatable sequence. Of course the catch-22 is that to
determine if a sequence has 0 autocerrelation you have to look at it
forever.