Say we have a space of $n\times m$ pixels. We want to sample each pixel as either $+1$ or $-1$ while all other pixels are valued at $0$. Of note, the location of each pixel is important, and there are dependencies across pixels, so each pixel needs to be sampled independently.
To build some intuition, the simplest solution to this problem would be to set all pixels to $0$, then value our first pixel a $-1$, followed by $+1$. We would then move to a second pixel and repeat. This would fully sample the set.
However, we can undoubtedly speed up the process by providing combinations of pixels. What is the minimum number of groups of pixels needed to sample the entire space? How might these arrangements of pixels be distributed?
Bonus: If we set the requirement that, when a pixel is valued at $(-1,+1)$, all pixels in radius $r$ must be valued at $0$, how many combinations do we encounter, and what is their distribution?