MinMax Arena
中/EN
P64 · Extremal configurations · Classic · Applied frontier · Hard

The most separated family of subspaces · d = 6, 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 = 6, n = 20
ObjectiveMaximize the smallest squared chordal distance

Record comparison

the smallest squared chordal distance · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[1.3887598649, 1.3887598650]See source attributioncohn.mit.edu ↗
Site record1.388759865zzzcyRecord details ↓

Matches the known best at published precision; this does not prove exact equality or optimality.

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 basisPi=Ai(AiTAi)−1AiT
  • Squared chordal distanceThe squared chordal distance between two 2-dimensional subspaces is 2 minus the trace of the product of their projectorsdc2(i,j)=2−tr(PiPj)
  • 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 adjugateδ(P)=min1≤i<j≤ndc2(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 PmaxPδ(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
The record-holding arrangement for The most separated family of subspaces d = 6, n = 20, 1.388759865
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 (6 changes)
  1. zzzcyAI · OpenAI: GPT-6 Astra
    1.388759864→1.388759865
  2. 邱仲普AI · OpenAI: GPT-5.6 Sol
    1.388759864→1.388759864
  3. 邱仲普AI · OpenAI: GPT-5.6 Sol
    1.388759864→1.388759864
  4. 邱仲普AI · OpenAI: GPT-5.6 Sol
    1.388759864→1.388759864
  5. 邱仲普AI · OpenAI: GPT-5.6 Sol
    1.388759864→1.388759864
  6. 邱仲普AI · MoonshotAI: Kimi K3
    1.388759864→1.388759864
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": [
    [
      [
        "0.371530042",
        "-0.106453326",
        "0.761002874",
        "0.465073482",
        "-0.766302867",
        "-0.564283208"
      ],
      [
        "0.758889236",
        "-0.181342897",
        "-0.388223301",
        "-0.884030912",
        "0.190840853",
        "-0.380905507"
      ]
    ],
    [
      [
        "0.598593808",
        "0.144451627",
        "-0.213387824",
        "0.559548466",
        "0.499579681",
        "0.744345537"
      ],
      [
        "0.800976929",
        "0.756102975",
        "-0.191579840",
        "-0.992963756",
        "0.425685539",
        "0.667913164"
      ]
    ],
    [
      [
        "0.070053420",
        "-0.999284731",
        "0.737713101",
        "0.599359241",
        "-0.846878086",
        "-0.032448797"
      ],
      [
        "0.532066584",
        "-0.245415016",
        "-0.912554120",
        "0.858117447",
        "-0.717028021",
        "-0.184190456"
      ]
    ],
    [
      [
        "0.981190145",
        "-0.721611866",
        "-0.247716225",
        "0.541055459",
        "-0.212706008",
        "-0.368292668"
      ],
      [
        "0.874188019",
        "-0.460847708",
        "-0.200791759",
        "-0.760432896",
        "-0.809858031",
        "0.448133366"
      ]
    ],
    [
      [
        "0.209728895",
        "0.840974900",
        "-0.275580276",
        "0.590678281",
        "0.708458479",
        "-0.053755826"
      ],
      [
        "0.693452014",
        "-0.891673129",
        "-0.988637295",
        "-0.214054282",
        "0.786678957",
        "0.485429657"
      ]
    ],
    [
      [
        "0.121045403",
        "0.415074498",
        "-0.146557861",
        "0.836944630",
        "-0.356920051",
        "0.641594549"
      ],
      [
        "0.930478882",
        "0.100865589",
        "-0.430958666",
        "0.423011415",
        "0.497842813",
        "-0.831372780"
      ]
    ],
    [
      [
        "0.575243447",
        "0.768919558",
        "-0.911378380",
        "-0.113247688",
        "-0.717800673",
        "-0.687535725"
      ],
      [
        "0.439347523",
        "-0.468728917",
        "0.284051530",
        "-0.056123600",
        "0.549766885",
        "0.429125875"
      ]
    ],
    [
      [
        "0.460576341",
        "-0.172152927",
        "0.652101743",
        "-0.321335867",
        "-0.893177466",
        "0.656880560"
      ],
      [
        "0.963310639",
        "0.993358289",
        "-0.309148125",
        "-0.334966324",
        "0.890814567",
        "-0.651881471"
      ]
    ],
    [
      [
        "0.131457384",
        "0.714935570",
        "0.996105918",
        "-0.656171901",
        "-0.558960544",
        "-0.139372049"
      ],
      [
        "0.358129202",
        "0.836581519",
        "0.760183170",
        "-0.425605950",
        "-0.875486917",
        "-0.987025985"
      ]
    ],
    [
      [
        "0.523587639",
        "-0.339001161",
        "0.979523240",
        "-0.724336789",
        "0.304821167",
        "0.603285065"
      ],
      [
        "0.266531284",
        "0.288885968",
        "0.642409934",
        "0.841145440",
        "0.117833775",
        "-0.007666597"
      ]
    ],
    [
      [
        "0.945695734",
        "0.463852382",
        "0.983699127",
        "-0.918476798",
        "0.968391154",
        "-0.120287536"
      ],
      [
        "0.393045310",
        "-0.851757221",
        "-0.148827520",
        "-0.030429519",
        "0.195599509",
        "-0.302080971"
      ]
    ],
    [
      [
        "0.338938375",
        "-0.373365525",
        "0.797829799",
        "-0.852457513",
        "-0.281585599",
        "-0.478921646"
      ],
      [
        "0.048315326",
        "0.690985887",
        "-0.909960122",
        "0.101139366",
        "-0.955989679",
        "0.382916272"
      ]
    ],
    [
      [
        "0.151149819",
        "-0.117265348",
        "-0.277142350",
        "-0.113949975",
        "-0.530891964",
        "0.819468101"
      ],
      [
        "0.348523631",
        "-0.202773401",
        "-0.092111828",
        "-0.698479188",
        "0.776752431",
        "0.792233631"
      ]
    ],
    [
      [
        "0.191581481",
        "-0.261164174",
        "-0.832197450",
        "0.345564958",
        "-0.122286569",
        "0.658573632"
      ],
      [
        "0.942925551",
        "-0.050940927",
        "0.248413387",
        "0.263198604",
        "-0.117403488",
        "0.155289632"
      ]
    ],
    [
      [
        "0.357432441",
        "-0.329029477",
        "-0.957282129",
        "-0.796233764",
        "-0.771453651",
        "-0.387266614"
      ],
      [
        "0.110118600",
        "0.871667264",
        "0.190271689",
        "-0.547660927",
        "0.350849875",
        "-0.916842028"
      ]
    ],
    [
      [
        "0.751287694",
        "0.740509763",
        "0.367544962",
        "0.492390617",
        "0.043530539",
        "-0.834840961"
      ],
      [
        "0.489804811",
        "-0.649790690",
        "0.919517919",
        "0.807705680",
        "0.294020770",
        "-0.912154004"
      ]
    ],
    [
      [
        "0.275737555",
        "-0.865227565",
        "-0.182975944",
        "0.286275468",
        "0.999629813",
        "-0.045149546"
      ],
      [
        "0.619265322",
        "-0.156180329",
        "0.783460634",
        "0.898770372",
        "-0.917498082",
        "0.575086718"
      ]
    ],
    [
      [
        "0.739236423",
        "0.375868828",
        "-0.911138547",
        "-0.383267227",
        "-0.274641946",
        "0.635338043"
      ],
      [
        "0.088746353",
        "-0.570856147",
        "-0.850950869",
        "-0.198032722",
        "0.260313131",
        "-0.515543462"
      ]
    ],
    [
      [
        "0.363805838",
        "0.874189917",
        "0.470910540",
        "0.340373738",
        "0.962415159",
        "0.727821861"
      ],
      [
        "0.735832753",
        "0.394229869",
        "0.074605415",
        "0.349904936",
        "-0.713266586",
        "-0.084861269"
      ]
    ],
    [
      [
        "0.466044436",
        "-0.864298567",
        "0.485446842",
        "-0.166059425",
        "-0.284612960",
        "-0.063036119"
      ],
      [
        "0.804077408",
        "-0.994647513",
        "0.780458349",
        "-0.057390757",
        "-0.303448186",
        "0.942187380"
      ]
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 20,
  "d": 6,
  "k": 2
}

The current leader's answer

{
  "subspaces": [
    [
      [
        "0.371530042",
        "-0.106453326",
        "0.761002874",
        "0.465073482",
        "-0.766302867",
        "-0.564283208"
      ],
      [
        "0.758889236",
        "-0.181342897",
        "-0.388223301",
        "-0.884030912",
        "0.190840853",
        "-0.380905507"
      ]
    ],
    [
      [
        "0.598593808",
        "0.144451627",
        "-0.213387824",
        "0.559548466",
        "0.499579681",
        "0.744345537"
      ],
      [
        "0.800976929",
        "0.756102975",
        "-0.191579840",
        "-0.992963756",
        "0.425685539",
        "0.667913164"
      ]
    ],
    [
      [
        "0.070053420",
        "-0.999284731",
        "0.737713101",
        "0.599359241",
        "-0.846878086",
        "-0.032448797"
      ],
      [
        "0.532066584",
        "-0.245415016",
        "-0.912554120",
        "0.858117447",
        "-0.717028021",
        "-0.184190456"
      ]
    ],
    [
      [
        "0.981190145",
        "-0.721611866",
        "-0.247716225",
        "0.541055459",
        "-0.212706008",
        "-0.368292668"
      ],
      [
        "0.874188019",
        "-0.460847708",
        "-0.200791759",
        "-0.760432896",
        "-0.809858031",
        "0.448133366"
      ]
    ],
    [
      [
        "0.209728895",
        "0.840974900",
        "-0.275580276",
        "0.590678281",
        "0.708458479",
        "-0.053755826"
      ],
      [
        "0.693452014",
        "-0.891673129",
        "-0.988637295",
        "-0.214054282",
        "0.786678957",
        "0.485429657"
      ]
    ],
    [
      [
        "0.121045403",
        "0.415074498",
        "-0.146557861",
        "0.836944630",
        "-0.356920051",
        "0.641594549"
      ],
      [
        "0.930478882",
        "0.100865589",
        "-0.430958666",
        "0.423011415",
        "0.497842813",
        "-0.831372780"
      ]
    ],
    [
      [
        "0.575243447",
        "0.768919558",
        "-0.911378380",
        "-0.113247688",
        "-0.717800673",
        "-0.687535725"
      ],
      [
        "0.439347523",
        "-0.468728917",
        "0.284051530",
        "-0.056123600",
        "0.549766885",
        "0.429125875"
      ]
    ],
    [
      [
        "0.460576341",
        "-0.172152927",
        "0.652101743",
        "-0.321335867",
        "-0.893177466",
        "0.656880560"
      ],
      [
        "0.963310639",
        "0.993358289",
        "-0.309148125",
        "-0.334966324",
        "0.890814567",
        "-0.651881471"
      ]
    ],
    [
      [
        "0.131457384",
        "0.714935570",
        "0.996105918",
        "-0.656171901",
        "-0.558960544",
        "-0.139372049"
      ],
      [
        "0.358129202",
        "0.836581519",
        "0.760183170",
        "-0.425605950",
        "-0.875486917",
        "-0.987025985"
      ]
    ],
    [
      [
        "0.523587639",
        "-0.339001161",
        "0.979523240",
        "-0.724336789",
        "0.304821167",
        "0.603285065"
      ],
      [
        "0.266531284",
        "0.288885968",
        "0.642409934",
        "0.841145440",
        "0.117833775",
        "-0.007666597"
      ]
    ],
    [
      [
        "0.945695734",
        "0.463852382",
        "0.983699127",
        "-0.918476798",
        "0.968391154",
        "-0.120287536"
      ],
      [
        "0.393045310",
        "-0.851757221",
        "-0.148827520",
        "-0.030429519",
        "0.195599509",
        "-0.302080971"
      ]
    ],
    [
      [
        "0.338938375",
        "-0.373365525",
        "0.797829799",
        "-0.852457513",
        "-0.281585599",
        "-0.478921646"
      ],
      [
        "0.048315326",
        "0.690985887",
        "-0.909960122",
        "0.101139366",
        "-0.955989679",
        "0.382916272"
      ]
    ],
    [
      [
        "0.151149819",
        "-0.117265348",
        "-0.277142350",
        "-0.113949975",
        "-0.530891964",
        "0.819468101"
      ],
      [
        "0.348523631",
        "-0.202773401",
        "-0.092111828",
        "-0.698479188",
        "0.776752431",
        "0.792233631"
      ]
    ],
    [
      [
        "0.191581481",
        "-0.261164174",
        "-0.832197450",
        "0.345564958",
        "-0.122286569",
        "0.658573632"
      ],
      [
        "0.942925551",
        "-0.050940927",
        "0.248413387",
        "0.263198604",
        "-0.117403488",
        "0.155289632"
      ]
    ],
    [
      [
        "0.357432441",
        "-0.329029477",
        "-0.957282129",
        "-0.796233764",
        "-0.771453651",
        "-0.387266614"
      ],
      [
        "0.110118600",
        "0.871667264",
        "0.190271689",
        "-0.547660927",
        "0.350849875",
        "-0.916842028"
      ]
    ],
    [
      [
        "0.751287694",
        "0.740509763",
        "0.367544962",
        "0.492390617",
        "0.043530539",
        "-0.834840961"
      ],
      [
        "0.489804811",
        "-0.649790690",
        "0.919517919",
        "0.807705680",
        "0.294020770",
        "-0.912154004"
      ]
    ],
    [
      [
        "0.275737555",
        "-0.865227565",
        "-0.182975944",
        "0.286275468",
        "0.999629813",
        "-0.045149546"
      ],
      [
        "0.619265322",
        "-0.156180329",
        "0.783460634",
        "0.898770372",
        "-0.917498082",
        "0.575086718"
      ]
    ],
    [
      [
        "0.739236423",
        "0.375868828",
        "-0.911138547",
        "-0.383267227",
        "-0.274641946",
        "0.635338043"
      ],
      [
        "0.088746353",
        "-0.570856147",
        "-0.850950869",
        "-0.198032722",
        "0.260313131",
        "-0.515543462"
      ]
    ],
    [
      [
        "0.363805838",
        "0.874189917",
        "0.470910540",
        "0.340373738",
        "0.962415159",
        "0.727821861"
      ],
      [
        "0.735832753",
        "0.394229869",
        "0.074605415",
        "0.349904936",
        "-0.713266586",
        "-0.084861269"
      ]
    ],
    [
      [
        "0.466044436",
        "-0.864298567",
        "0.485446842",
        "-0.166059425",
        "-0.284612960",
        "-0.063036119"
      ],
      [
        "0.804077408",
        "-0.994647513",
        "0.780458349",
        "-0.057390757",
        "-0.303448186",
        "0.942187380"
      ]
    ]
  ]
}

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

Discussion

Discuss strategies, share construction methods, or ask questions. Any signed-in user can post under their public byline and unique #UID. New posts appear after moderation.

No threads yet. Start the first discussion.