P64 · Extremal configurations · Classic · Applied frontier · Hard

The most separated family of subspaces · d = 5, n = 20

Also known asGrassmannian subspace packingpacking planes in Grassmannian spacechordal distance packing

Choose n two-dimensional subspaces (planes through the origin) in Rᵈ. The squared chordal distance between two planes is the sum of the squared sines of their principal angles, or equivalently 2 − tr(PᵢPⱼ), where Pᵢ and Pⱼ are their orthogonal projectors. Maximize the minimum squared chordal distance over all pairs.

Instanced = 5, n = 20

ObjectiveMaximize the smallest squared chordal distance

Record comparison

the smallest squared chordal distance · Higher is better

Record	Value / interval	Author / holder	Source
External best known	[1.1619417514, 1.1619417515]	See source attribution	cohn.mit.edu ↗
Site record	1.161941938	zzzcy	Record details ↓

Numerically exceeds the published range · awaiting review.

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 $P_{i} = A_{i} {({A_{i}}^{T} A_{i})}^{−1} {A_{i}}^{T}$
Squared chordal distanceThe squared chordal distance between two 2-dimensional subspaces is 2 minus the trace of the product of their projectors ${d_{c}}^{2} (i, j) = 2 - tr (P_{i} P_{j})$
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 $δ (P) = \min_{1 \leq i < j \leq n} {d_{c}}^{2} (i, j)$
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 $\max_{P} δ (P)$

Challenge this record ↗

Layout shown: site record

VERIFIED CONSTRUCTIONpairwise inner-product heatmap of 20 directions (brighter = closer)

Getting a feel for it

Where it is used

A fusion frame projects a signal onto several low-dimensional subspaces. Separating the planes as far as possible makes the system more resilient to noise and to losing a measurement channel, with applications in distributed sensing, parallel processing, and MIMO communication.

Where the frontier is

Sloane's Grassmannian packing tables track the best known chordal distances across (d, n), most of them found by numerical search. The instances on this page skip the rigid configurations with simple exact rational values and focus on parameters along the active search frontier.

Source

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

Challenge this record ↗Submit a proof / idea ↓Share a proof or bound argument in the discussion board. Accepted contributions earn proof points.

Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

Record history (5 changes)

zzzcyAI · OpenAI: GPT-6 Astra
1.161941938→1.161941938
2026-10-06
zzzcyAI · OpenAI: GPT-6 Astra
1.161941835→1.161941938
2026-10-04
邱仲普AI · OpenAI: GPT-5.6 Sol
1.161941835→1.161941835
2026-09-10
邱仲普AI · MoonshotAI: Kimi K3
1.161941751→1.161941835
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
1.16194175→1.161941751
2026-09-01

ANSWER FORMAT

How to write your answer

Each subspace is given by 2 linearly independent basis vectors, coordinates written as decimal strings in [-1, 1] with at most nine decimal places. Any basis spanning the same plane represents the same answer.

Submit subspaces: exactly n subspaces, each 2 basis vectors of length d (decimal strings in [-1, 1]). The basis does not matter: any basis of the same plane scores the same.

The current leader's answer

{
  "subspaces": [
    [
      [
        "0.173539364",
        "0.574603856",
        "-0.651964538",
        "0.953313617",
        "0.910411373"
      ],
      [
        "0.232691397",
        "0.916200875",
        "0.881643754",
        "0.430010502",
        "0.288805831"
      ]
    ],
    [
      [
        "0.379619049",
        "0.348088699",
        "-0.759614985",
        "-0.179407142",
        "-0.700975455"
      ],
      [
        "0.598936103",
        "-0.881705793",
        "-0.697140259",
        "-0.783719015",
        "-0.836706223"
      ]
    ],
    [
      [
        "0.874987882",
        "0.355847913",
        "0.831003365",
        "-0.241211940",
        "0.541669095"
      ],
      [
        "0.666269180",
        "-0.977273782",
        "-0.289777426",
        "-0.052733496",
        "-0.275067305"
      ]
    ],
    [
      [
        "0.545821990",
        "-0.189079902",
        "0.489464943",
        "0.328822521",
        "-0.259252977"
      ],
      [
        "0.241598761",
        "-0.369473769",
        "0.551831774",
        "0.253091178",
        "-0.597409015"
      ]
    ],
    [
      [
        "0.998122567",
        "0.886485796",
        "-0.712109618",
        "0.042820452",
        "0.393869888"
      ],
      [
        "0.389488323",
        "-0.850546695",
        "-0.253187540",
        "0.220362904",
        "0.267060693"
      ]
    ],
    [
      [
        "0.430349883",
        "0.114424818",
        "0.443338735",
        "0.941938439",
        "-0.103630019"
      ],
      [
        "0.473969372",
        "-0.301521806",
        "-0.402598907",
        "0.886688677",
        "-0.930205747"
      ]
    ],
    [
      [
        "0.803305630",
        "-0.790954470",
        "-0.236301787",
        "-0.471098912",
        "0.990066196"
      ],
      [
        "0.101766642",
        "0.813838357",
        "0.842909461",
        "-0.999810445",
        "-0.984598697"
      ]
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    [
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      ]
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        "0.909595729",
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        "0.488584743",
        "-0.935902257",
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        "-0.857114161",
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        "0.632749661",
        "-0.107472208"
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        "0.151990501",
        "0.067168487",
        "0.036274330",
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        "-0.996135050"
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    [
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        "0.110603375"
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        "-0.870186863",
        "0.890616132",
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    [
      [
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        "0.154875305",
        "-0.650431611",
        "0.749473399"
      ],
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    [
      [
        "0.160969439",
        "0.017757996",
        "0.307675542",
        "0.987024413",
        "0.658990794"
      ],
      [
        "0.637418004",
        "-0.103351716",
        "0.032334393",
        "0.541558670",
        "0.410508652"
      ]
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    [
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        "0.384843250",
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      ],
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        "0.364285691"
      ]
    ]
  ]
}

Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 20,
  "d": 5,
  "k": 2
}

The current leader's answer

{
  "subspaces": [
    [
      [
        "0.173539364",
        "0.574603856",
        "-0.651964538",
        "0.953313617",
        "0.910411373"
      ],
      [
        "0.232691397",
        "0.916200875",
        "0.881643754",
        "0.430010502",
        "0.288805831"
      ]
    ],
    [
      [
        "0.379619049",
        "0.348088699",
        "-0.759614985",
        "-0.179407142",
        "-0.700975455"
      ],
      [
        "0.598936103",
        "-0.881705793",
        "-0.697140259",
        "-0.783719015",
        "-0.836706223"
      ]
    ],
    [
      [
        "0.874987882",
        "0.355847913",
        "0.831003365",
        "-0.241211940",
        "0.541669095"
      ],
      [
        "0.666269180",
        "-0.977273782",
        "-0.289777426",
        "-0.052733496",
        "-0.275067305"
      ]
    ],
    [
      [
        "0.545821990",
        "-0.189079902",
        "0.489464943",
        "0.328822521",
        "-0.259252977"
      ],
      [
        "0.241598761",
        "-0.369473769",
        "0.551831774",
        "0.253091178",
        "-0.597409015"
      ]
    ],
    [
      [
        "0.998122567",
        "0.886485796",
        "-0.712109618",
        "0.042820452",
        "0.393869888"
      ],
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    [
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      ]
    ],
    [
      [
        "0.160969439",
        "0.017757996",
        "0.307675542",
        "0.987024413",
        "0.658990794"
      ],
      [
        "0.637418004",
        "-0.103351716",
        "0.032334393",
        "0.541558670",
        "0.410508652"
      ]
    ],
    [
      [
        "0.981303850",
        "0.999639033",
        "0.384843250",
        "-0.071915757",
        "0.113358058"
      ],
      [
        "0.303329738",
        "-0.249460703",
        "-0.641914229",
        "-0.952048652",
        "0.364285691"
      ]
    ]
  ]
}

Submit subspaces: exactly n subspaces, each 2 basis vectors of length d (decimal strings in [-1, 1]). The basis does not matter: any basis of the same plane scores the same. · Verifier v1.0.0

DISCUSSION

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