I am looking for an $\approx12\times12$ rectangle (small holes and small obtrusions are okay) made entirely of cube net hexominos.
It is my understanding that perfect rectangles, in general, are not possible using the set of $35$ hexominos. I am aware that restricting use to only cube net polyominos exacerbates this limitation.
How close to a rectangle of order $12\times12$ can one get using only cube net hexominos, where the measure of distance to a $12\times12$ rectangle is the number of 'single-square' deletions/additions that would be necessary to delete spurious edges or fill holes.
I would also be interested in known tilings of cube net hexominos, no matter how close they are to a $12\times12$ rectangle.
It's possible this problem may be solvable using computer search. I'm working on that right now.
There is no nice solution with these pieces. Use Burr Tools to find more solutions.