Aieee! I have an algorithm that finds
K-uniform 2^K-periodic binary sequences for
K = 1..5, but not 6 and 7, and probably
nothing higher than that. I had verified
it up to K = 4 by hand and assumed it
worked for all K.
Damndamndamn.
So I'm not entirely sure if a
6-uniform 64-periodic sequence (in
particular) exists.