For those who thought the 21q/16h above was interesting, here is another 21q/16h which preserves everything on the cube but the U and F faces:The only dedge flip algorithm I can recall which did need to destroy additional dedges to be briefer (not double parity, only OLL) was the one I showed you before:
r U2 r' U2 r D2 r D2 r' F L' F' U' r' D2 r
That DOES fit the "mess up more, less moves" idea, though definitely messes up too much.
In WCA notation:
Rw U2 Rw' U2 Rw D2 r D2 l' U L' U' B' l' U2 l x'
Note 1: This algorithm must be performed exactly as shown in order to preserve the most. If all slice turns are wide, then it will scramble the reduced 4X4X4 just as much as the other one. If all slice turns are converted to wide except for the bold r turn, then still a 3X3X3 block is preserved:
Rw U2 Rw' U2 Rw D2 r D2 Lw' U L' U' B' Lw' U2 Lw x'
Note 2: This is not a double parity algorithm. The single slice version is two 2-cycles and a 4-cycle of wing edge pieces, hence giving the overall effect on a 4X4X4 just a single edge flip. (The other algorithm I previously gave out was two 4-cycles and a 2-cycle).
r U2 r' U2 r D2 r D2 l' U L' U' B' l' U2 l
U2 B U L U' M
As a matter of fact, by the cube generated from the algorithm above, it is evident that I used the front-left F2L slot to create a sub-25q algorithm. In addition, the 21q/16h is just the inner-layer workings of this overall 4-cycle+ two 2-cycle algorithm. The rest of the moves are neglected because they do not need to be executed to help break the odd permutation in the inner orbit, nor restore the centers and repair up the dedges.
These algorithms are NOT for odd cubes. Only for the 4X4X4 and larger even cubes which are reduced to 4X4X4s.
It obviously can't be used for speedsolving, but I thought I would share it with everyone to show that 21qs like this do not need to scramble the reduced 4X4X4 completely. It's pretty cool to do a double parity and adjacent PLL parity in 29q though (I am referring to the full algorithm).