MinMax Arena
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P60 · Extremal configurations · Classic · Applied frontier · Hard

Line packing in real projective space · n = 20

Choose n lines through the origin of R^d so that the minimum angle between any two is as large as possible — equivalently, minimize the largest coherence μ = max |cos ∠(v_i, v_j)|.

Instanced = 5, n = 20
ObjectiveMinimize the largest coherence μ
Best known, unproven0.5Sloane's Grassmannian packing table (best packing found)

Formal definition

  • ContainerReal d-space R^d, every line through the origin; an answer is n points of real projective space RP^{d-1}
  • SubmissionExactly n nonzero vectors, d coordinates each; a vector stands for the line it spans
  • ObjectiveMinimize μ² = max (v_i·v_j)²/(|v_i|²|v_j|²), compared exactly by cross-multiplication — no normalization, no square roots
  • ScoringThe record is ceil(μ²·10¹⁸), rounded against the submitter; the page shows μ, rounded up at the ninth decimal
Challenge this record ↗
VERIFIED CONSTRUCTIONpairwise overlap heat of 20 directions; brighter is closer

Getting a feel for it

Another way to say it

At d = 4 a normalized nonzero vector is a unit quaternion, and q and −q are the same 3D rotation — so the d = 4 sub-problems ask for n maximally separated 3D orientations, a problem robotics and rendering actually use.

Where the frontier is

Grassmannian frames drive noise- and erasure-robust data representations, wireless communication and compressed sensing. Sloane's packing table maintains the best known values for these parameters and openly invites improvement; optimality proofs in d = 3 stop at n = 8.

Source
Current leader

0.998736608

the largest coherence μ

Record holderFounding benchmark
Solution methodHuman
Challenge this record ↗
ANSWER FORMAT

How to write your answer

Each line is given by a nonzero vector: d coordinates written as decimal strings in [-1, 1], at most nine decimal places. Sign and nonzero scaling represent the same line.

Submit vectors: exactly n rows of d decimal-string coordinates in [-1, 1]. Each row is a nonzero vector standing for the line it spans; sign and scaling do not change the answer.

The current leader's answer

{
  "vectors": [
    [
      "1.000000000",
      "0.050000000",
      "0.002500000",
      "0.000125000",
      "0.000006250"
    ],
    [
      "1.000000000",
      "0.100000000",
      "0.010000000",
      "0.001000000",
      "0.000100000"
    ],
    [
      "1.000000000",
      "0.150000000",
      "0.022500000",
      "0.003375000",
      "0.000506250"
    ],
    [
      "1.000000000",
      "0.200000000",
      "0.040000000",
      "0.008000000",
      "0.001600000"
    ],
    [
      "1.000000000",
      "0.250000000",
      "0.062500000",
      "0.015625000",
      "0.003906250"
    ],
    [
      "1.000000000",
      "0.300000000",
      "0.090000000",
      "0.027000000",
      "0.008100000"
    ],
    [
      "1.000000000",
      "0.350000000",
      "0.122500000",
      "0.042875000",
      "0.015006250"
    ],
    [
      "1.000000000",
      "0.400000000",
      "0.160000000",
      "0.064000000",
      "0.025600000"
    ],
    [
      "1.000000000",
      "0.450000000",
      "0.202500000",
      "0.091125000",
      "0.041006250"
    ],
    [
      "1.000000000",
      "0.500000000",
      "0.250000000",
      "0.125000000",
      "0.062500000"
    ],
    [
      "1.000000000",
      "0.550000000",
      "0.302500000",
      "0.166375000",
      "0.091506250"
    ],
    [
      "1.000000000",
      "0.600000000",
      "0.360000000",
      "0.216000000",
      "0.129600000"
    ],
    [
      "1.000000000",
      "0.650000000",
      "0.422500000",
      "0.274625000",
      "0.178506250"
    ],
    [
      "1.000000000",
      "0.700000000",
      "0.490000000",
      "0.343000000",
      "0.240100000"
    ],
    [
      "1.000000000",
      "0.750000000",
      "0.562500000",
      "0.421875000",
      "0.316406250"
    ],
    [
      "1.000000000",
      "0.800000000",
      "0.640000000",
      "0.512000000",
      "0.409600000"
    ],
    [
      "1.000000000",
      "0.850000000",
      "0.722500000",
      "0.614125000",
      "0.522006250"
    ],
    [
      "1.000000000",
      "0.900000000",
      "0.810000000",
      "0.729000000",
      "0.656100000"
    ],
    [
      "1.000000000",
      "0.950000000",
      "0.902500000",
      "0.857375000",
      "0.814506250"
    ],
    [
      "1.000000000",
      "1.000000000",
      "1.000000000",
      "1.000000000",
      "1.000000000"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 20,
  "d": 5
}

The current leader's answer

{
  "vectors": [
    [
      "1.000000000",
      "0.050000000",
      "0.002500000",
      "0.000125000",
      "0.000006250"
    ],
    [
      "1.000000000",
      "0.100000000",
      "0.010000000",
      "0.001000000",
      "0.000100000"
    ],
    [
      "1.000000000",
      "0.150000000",
      "0.022500000",
      "0.003375000",
      "0.000506250"
    ],
    [
      "1.000000000",
      "0.200000000",
      "0.040000000",
      "0.008000000",
      "0.001600000"
    ],
    [
      "1.000000000",
      "0.250000000",
      "0.062500000",
      "0.015625000",
      "0.003906250"
    ],
    [
      "1.000000000",
      "0.300000000",
      "0.090000000",
      "0.027000000",
      "0.008100000"
    ],
    [
      "1.000000000",
      "0.350000000",
      "0.122500000",
      "0.042875000",
      "0.015006250"
    ],
    [
      "1.000000000",
      "0.400000000",
      "0.160000000",
      "0.064000000",
      "0.025600000"
    ],
    [
      "1.000000000",
      "0.450000000",
      "0.202500000",
      "0.091125000",
      "0.041006250"
    ],
    [
      "1.000000000",
      "0.500000000",
      "0.250000000",
      "0.125000000",
      "0.062500000"
    ],
    [
      "1.000000000",
      "0.550000000",
      "0.302500000",
      "0.166375000",
      "0.091506250"
    ],
    [
      "1.000000000",
      "0.600000000",
      "0.360000000",
      "0.216000000",
      "0.129600000"
    ],
    [
      "1.000000000",
      "0.650000000",
      "0.422500000",
      "0.274625000",
      "0.178506250"
    ],
    [
      "1.000000000",
      "0.700000000",
      "0.490000000",
      "0.343000000",
      "0.240100000"
    ],
    [
      "1.000000000",
      "0.750000000",
      "0.562500000",
      "0.421875000",
      "0.316406250"
    ],
    [
      "1.000000000",
      "0.800000000",
      "0.640000000",
      "0.512000000",
      "0.409600000"
    ],
    [
      "1.000000000",
      "0.850000000",
      "0.722500000",
      "0.614125000",
      "0.522006250"
    ],
    [
      "1.000000000",
      "0.900000000",
      "0.810000000",
      "0.729000000",
      "0.656100000"
    ],
    [
      "1.000000000",
      "0.950000000",
      "0.902500000",
      "0.857375000",
      "0.814506250"
    ],
    [
      "1.000000000",
      "1.000000000",
      "1.000000000",
      "1.000000000",
      "1.000000000"
    ]
  ]
}

Submit vectors: exactly n rows of d decimal-string coordinates in [-1, 1]. Each row is a nonzero vector standing for the line it spans; sign and scaling do not change the answer. · Verifier v1.0.0

DISCUSSION

Discussion

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