# sequentially sampling an n-dimensional space

lets say I have a 5 dimensional grid where each dimension has 10 points. There are 100000 combinations. Let's say I want a subset of 10000, is there a deterministic algorithm that will choose a set of points that "covers" the region the best and converges to the limiting case of choosing every point?

If I had a weighting function for each point, would there be a deterministic algorithm that would tend to choose points that are equidistant in a weighting sense as well?

I can think of some possible ways in lower dimension, just wondering if there is a well defined solution (area of math) for these types of problems that extends to higher dimensions

I've tried searching sampling methods, multi-dimensional sampling and the like, but no luck

Let's talk about the uniform sampling case. Suppose you define some spatially regular ordering for the points on the grid. For the 5 dimensional points, say the first coordinate increases the slowest, then the second increases the next slowest, and so on, and the last coordinate increases the fastest. Denote the ordered set $p_i$ for $i=0,\ldots,N-1$ where $N=100000$ is the total number of points. Let $n$ be the number of points we want to sample. If you take the set of points $${ p_j: j = (a+iM) \mod N }$$ for arbitrary $a$ and $i=0,\ldots,n$ where $M$ and $N$ are coprime, then the $p_j$ should not repeat until $n = N$. If you choose $a$ and $M$ reasonably ($M$ being quite a bit larger than $N$), then these points will also be relatively uniformly distributed, but perhaps with too much structure and no guarantee that they have the blue noise property.
For the weighted case, you are essentially asking for a combinatorial optimization, so I doubt there would be efficient ways of solving it. On the other hand, your problem size of $10^5$ is not that big.
You enumerate your grid points along a space-filling curve. If you then want $M$ out of $N$ points, choose the points with indices $k\frac{N}{M}$, where $k=1\dots M$.