Codebook packing in complex projective space
Choose n nonzero vectors in C^d — n points of complex projective space — minimizing the largest normalized Hermitian overlap μ = max |⟨z_i, z_j⟩| / (|z_i||z_j|).
Formal definition
- ContainerComplex d-space C^d; an answer is n points of complex projective space CP^{d-1}
- SubmissionExactly n nonzero complex vectors, each d pairs [re, im]
- ObjectiveMinimize μ² = max |⟨z_i,z_j⟩|²/(|z_i|²|z_j|²); moduli and norms squared are rational, compared exactly by cross-multiplication
- ScoringThe record is ceil(μ²·10¹⁸); the page shows μ, rounded up at the ninth decimal
Getting a feel for it
Where it is used
Complex projective codebooks are quantum measurements (the SIC-POVM family), communication codebooks, and noise-robust data representations. The further apart the codewords, the better they survive noise and erasures.
Where the frontier is
The Game of Sloanes is a public leader board of putatively optimal packings that explicitly invites improvement. Every sub-problem here is chosen from its still-open rows — the ones with a real gap between the best known value and the lower bound.
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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