I think partitions can get you more orthogonality than your specific example of overlapping orthogonal sets. Take n vectors and pack them into d dimensions in two ways:
A tegum product with k subspaces, giving (n/k) vectors per subspace and n^2*(1-1/k)orthogonal pairs.
(n/d) sets of vectors, each internally orthogonal but each overlapping with the others, giving n*d orthogonal pairs.
If d < n*(1-1/k) the tegum product buys you more orthogonal pairs. If n > d then picking large k (so low-dimensional spaces) makes the tegum product preferred.
This doesn’t mean there isn’t some other arrangement that does better though...
Oh yes you’re totally right.
I think partitions can get you more orthogonality than your specific example of overlapping orthogonal sets. Take n vectors and pack them into d dimensions in two ways:
A tegum product with k subspaces, giving (n/k) vectors per subspace and n^2*(1-1/k)orthogonal pairs.
(n/d) sets of vectors, each internally orthogonal but each overlapping with the others, giving n*d orthogonal pairs.
If d < n*(1-1/k) the tegum product buys you more orthogonal pairs. If n > d then picking large k (so low-dimensional spaces) makes the tegum product preferred.
This doesn’t mean there isn’t some other arrangement that does better though...
Yeah, agreed that’s not an optimal arrangement, that was just a proof of concept for ’non tegum things can get a lot of orthogonality