The most separated family of subspaces

Formal definition

• ContainerReal d-space Rᵈ; an answer consists of n points in the Grassmannian G(d, 2), namely n planes through the origin
• SubmissionExactly n subspaces, each 2 basis vectors; the basis vectors must be linearly independent
• ProjectorThe two columns of Aᵢ are the submitted basis vectors of subspace i; Pᵢ is the orthogonal projector and does not depend on the chosen basis$$P_i=A_i{\left({A_i}^T{A}_i\right)}^{\mathrm{-1}}{A_i}^T$$
• Squared chordal distanceThe squared chordal distance between two 2-dimensional subspaces is 2 minus the trace of the product of their projectors$${d_c}^2(i,j)=2-\mathrm{tr}\left(P_i{P}_j\right)$$
• Scoreδ(P) is the smallest squared chordal distance over all pairs of subspaces; the verifier constructs the projectors in exact rational arithmetic via the 2×2 adjugate$$\delta \left(P\right)=\underset{1\le i<j\le n}{\min }{d_c}^2(i,j)$$
• ScoringThe record is stored as floor(min squared chordal distance · 10¹⁸); the page displays the squared chordal distance, rounded down at the ninth decimal
• ObjectiveMake δ(P) as large as possible over all legal families of subspaces P$$\underset{P}{\max }\delta \left(P\right)$$