The most separated family of subspaces
Choose n two-dimensional subspaces of R^d. The squared chordal distance of two planes is the sum of the squared sines of their principal angles — equivalently 2 − tr(P_iP_j) with P the orthogonal projectors. Maximize the smallest squared chordal distance over all pairs.
Formal definition
- ContainerReal d-space R^d; an answer is n points of the Grassmannian G(d, 2) — n planes through the origin
- SubmissionExactly n subspaces, each 2 basis vectors; the basis vectors must be linearly independent
- ObjectiveMaximize min 2 − tr(P_iP_j); projectors are built exactly through the 2×2 adjugate, rational throughout
- ScoringThe record is floor(min squared chordal distance · 10¹⁸), rounded against the submitter; the page shows the squared distance, rounded down at the ninth decimal
Getting a feel for it
Where it is used
A fusion frame projects a signal onto several low-dimensional subspaces. The further apart the planes, the more robust the system is to noise and to losing one of the measurements — distributed sensing, parallel processing, MIMO communication.
Where the frontier is
Sloane's Grassmannian packing tables maintain the best known chordal values for these parameters, most of them products of large searches; the sub-problems here deliberately skip the table's exactly-rational rigid plateaus and sit on the genuine search frontier.
SourceCurrent best solutions by n
Each n is an independent record with a page of its own. Open any of them to inspect the current construction, then challenge it.
Discussion (0) ↓Discussion
Talk strategy, share methods, ask why you are stuck. Posts carry your public byline, the same name your records use; the #number after it is the account's signup ordinal, so a name cannot be worn by someone else. The floor is earned: break a record once, anywhere, and it is yours for good. New posts appear after an automated review.
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