P64 · 极值构型 · 经典问题 · 应用前沿 · 难

最分离的子空间族 · d = 6, n = 17

文献中也称Grassmannian subspace packingpacking planes in Grassmannian spacechordal distance packing

在 Rᵈ 中选 n 个过原点的二维子空间（平面）。两个平面的弦距平方等于它们两个主夹角正弦值的平方和，也等于 2 − tr(PᵢPⱼ)（其中 Pᵢ、Pⱼ 是各自的正交投影矩阵）。让所有平面对中最小的弦距平方尽可能大。

子题d = 6, n = 17

目标最大化最小弦距平方

纪录对比

最小弦距平方 · 越大越好

纪录	数值 / 区间	作者 / 持有人	来源
外部已知最好	[1.4127250337, 1.4127250338]	见来源署名	cohn.mit.edu ↗
本站纪录	1.412725033	zzzcy	纪录详情 ↓

本站纪录尚未确认达到或超越此参考纪录。

严格定义

容器d 维实空间 Rᵈ；答案是 Grassmann 流形 G(d, 2) 中的 n 个点，即 n 个过原点的二维平面
提交恰好 n 个子空间，每个 2 个基向量；基向量必须线性无关
投影矩阵Aᵢ 的两列是第 i 个子空间的两个基向量；Pᵢ 是与基的选择无关的正交投影矩阵 $P_{i} = A_{i} {({A_{i}}^{T} A_{i})}^{−1} {A_{i}}^{T}$
弦距平方两个二维子空间的弦距平方等于 2 减去投影矩阵乘积的迹 ${d_{c}}^{2} (i, j) = 2 - tr (P_{i} P_{j})$
分数δ(P) 是所有子空间对中最小的弦距平方；验证器通过 2×2 伴随矩阵用精确有理数构造投影矩阵 $δ (P) = \min_{1 \leq i < j \leq n} {d_{c}}^{2} (i, j)$
计分纪录存为 floor(最小弦距平方 · 10¹⁸)；页面显示弦距平方，向下取整到第 9 位小数
目标在所有合法子空间族 P 中让 δ(P) 尽可能大 $\max_{P} δ (P)$

挑战这个纪录 ↗

当前展示：本站纪录

已验证构造17 个方向的两两重合度热图，越亮越接近

帮助理解

它用在哪里

Fusion 框架把高维信号分别投影到多个低维子空间上。各个平面之间分得越开，系统对测量噪声和单个通道丢失的容忍度就越高，常用于分布式传感、并行信号处理和 MIMO 通信。

前沿在哪里

Sloane 的 Grassmannian 打包表按弦距记录了各参数下的已知最好结果，其中大部分来自数值搜索。本页子题避开了表中那些恰好取到简单有理数的刚性构形，专门挑了仍有改进空间的搜索前沿参数。

查看来源

构型与历史

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者zzzcy

解题方式AI · OpenAI: GPT-6 Astra

挑战这个纪录 ↗提交证明 / 思路 ↓在讨论区分享最优性证明或关键界限思路，经审核采纳后可获证明分。

纪录保持者的求解笔记暂无求解笔记（点击展开）＋

纪录保持者暂未分享求解过程。

历史纪录（8 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
1.412725033→1.412725033
2026-10-06
zzzcyAI · OpenAI: GPT-6 Astra
1.412725033→1.412725033
2026-10-04
邱仲普AI · OpenAI: GPT-5.6 Sol
1.412725033→1.412725033
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.412725033→1.412725033
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.412725033→1.412725033
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.412725033→1.412725033
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.412725033→1.412725033
2026-09-10
邱仲普AI · MoonshotAI: Kimi K3
1.412725032→1.412725033
2026-09-04

ANSWER FORMAT

答案怎么写

每个子空间由 2 个线性无关的基向量给出，坐标写成 [-1, 1] 内的十进制字符串，最多九位小数。张成同一平面的任何基代表同一个答案。

提交 subspaces：恰好 n 个子空间，每个是 2 个长度为 d 的基向量（十进制字符串，[-1, 1]）。基的选择不影响分数：同一个平面的任何基得同一个分。

当前第一名的答案

{
  "subspaces": [
    [
      [
        "0.844636312",
        "0.603753075",
        "-0.642766479",
        "-0.358556719",
        "-0.422552694",
        "0.860417325"
      ],
      [
        "0.082706093",
        "0.243074760",
        "-0.760387750",
        "0.293714612",
        "-0.345545662",
        "-0.221777011"
      ]
    ],
    [
      [
        "0.594371253",
        "-0.308257693",
        "0.448811996",
        "0.881147349",
        "-0.480254929",
        "0.578821984"
      ],
      [
        "0.785087462",
        "0.190879211",
        "-0.991874283",
        "0.142884708",
        "0.386870026",
        "0.277601429"
      ]
    ],
    [
      [
        "0.173860579",
        "-0.253402902",
        "0.078671473",
        "0.765914443",
        "0.945685853",
        "-0.056507260"
      ],
      [
        "0.827793890",
        "0.049908919",
        "0.099489172",
        "-0.415254967",
        "-0.551358693",
        "0.217894395"
      ]
    ],
    [
      [
        "0.033854285",
        "0.757945275",
        "0.238949304",
        "0.438245108",
        "-0.334446893",
        "0.429302173"
      ],
      [
        "0.204182853",
        "0.929988560",
        "-0.350059161",
        "-0.848899234",
        "-0.437667481",
        "0.296452200"
      ]
    ],
    [
      [
        "0.360420465",
        "0.805470901",
        "-0.149595759",
        "-0.534757414",
        "-0.455359142",
        "-0.830880569"
      ],
      [
        "0.437114881",
        "-0.119600487",
        "0.943855695",
        "0.105943474",
        "0.044285375",
        "-0.172589095"
      ]
    ],
    [
      [
        "0.554578364",
        "0.994254430",
        "-0.207607087",
        "-0.476824317",
        "0.833951004",
        "-0.910616681"
      ],
      [
        "0.130498709",
        "0.134857521",
        "0.359614099",
        "-0.819176124",
        "-0.188781866",
        "0.197245211"
      ]
    ],
    [
      [
        "0.207907733",
        "-0.071207230",
        "-0.050195652",
        "0.361311424",
        "0.629145577",
        "0.676341414"
      ],
      [
        "0.134928052",
        "0.654983616",
        "-0.164260034",
        "0.661474242",
        "-0.649906976",
        "-0.738640991"
      ]
    ],
    [
      [
        "0.380773321",
        "-0.230612485",
        "0.828823822",
        "-0.302137492",
        "-0.703337115",
        "0.116785538"
      ],
      [
        "0.164965248",
        "0.826698909",
        "0.552945624",
        "0.282394660",
        "0.402826360",
        "0.750848139"
      ]
    ],
    [
      [
        "0.630489438",
        "-0.574754713",
        "0.481340695",
        "0.308423087",
        "-0.380026497",
        "-0.284757812"
      ],
      [
        "0.857761883",
        "-0.467225664",
        "0.453747279",
        "-0.937809737",
        "0.540031263",
        "0.120052756"
      ]
    ],
    [
      [
        "0.677850118",
        "-0.942646756",
        "0.130539554",
        "0.165403325",
        "0.144472613",
        "0.866371766"
      ],
      [
        "0.390808129",
        "-0.431950745",
        "-0.363629095",
        "0.151008660",
        "0.147015335",
        "-0.976546017"
      ]
    ],
    [
      [
        "0.241987594",
        "-0.212666150",
        "-0.819999111",
        "0.371478787",
        "-0.317923899",
        "0.900352521"
      ],
      [
        "0.925001583",
        "-0.722520538",
        "-0.320205734",
        "-0.951005329",
        "-0.503615768",
        "-0.103984791"
      ]
    ],
    [
      [
        "0.799617216",
        "0.685518677",
        "-0.273510234",
        "0.188212646",
        "-0.749330893",
        "-0.153061269"
      ],
      [
        "0.514793507",
        "-0.966608151",
        "-0.103795319",
        "0.148429063",
        "-0.842036626",
        "0.192214407"
      ]
    ],
    [
      [
        "0.501700619",
        "-0.630635106",
        "-0.941520545",
        "0.562417799",
        "-0.470199155",
        "-0.320196694"
      ],
      [
        "0.234337612",
        "0.452882174",
        "0.883354908",
        "0.417236495",
        "0.441780915",
        "-0.138072359"
      ]
    ],
    [
      [
        "0.543895573",
        "-0.043651670",
        "-0.872613388",
        "-0.368633003",
        "-0.625896534",
        "-0.720562789"
      ],
      [
        "0.882390705",
        "-0.036946036",
        "-0.635323156",
        "-0.609106719",
        "0.677560181",
        "-0.695433859"
      ]
    ],
    [
      [
        "0.148506435",
        "0.114525850",
        "0.140279562",
        "-0.042276094",
        "-0.853059414",
        "0.517489010"
      ],
      [
        "0.182730986",
        "0.230320362",
        "0.621102229",
        "0.453308682",
        "-0.437048296",
        "-0.717226410"
      ]
    ],
    [
      [
        "0.049173651",
        "-0.804562703",
        "-0.651927470",
        "-0.028032235",
        "0.238543933",
        "0.273858848"
      ],
      [
        "0.881981975",
        "-0.514930328",
        "-0.686190501",
        "-0.391210656",
        "0.750021205",
        "0.611831926"
      ]
    ],
    [
      [
        "0.462942077",
        "-0.325939805",
        "0.360881381",
        "0.028648666",
        "0.243208775",
        "-0.686963356"
      ],
      [
        "0.633656189",
        "0.810905443",
        "-0.363646434",
        "0.280529830",
        "0.324692054",
        "0.254617542"
      ]
    ]
  ]
}

提交格式与技术细节需要编写程序或准备 JSON 答案时再查看＋

子题参数

{
  "n": 17,
  "d": 6,
  "k": 2
}

当前第一名的答案

{
  "subspaces": [
    [
      [
        "0.844636312",
        "0.603753075",
        "-0.642766479",
        "-0.358556719",
        "-0.422552694",
        "0.860417325"
      ],
      [
        "0.082706093",
        "0.243074760",
        "-0.760387750",
        "0.293714612",
        "-0.345545662",
        "-0.221777011"
      ]
    ],
    [
      [
        "0.594371253",
        "-0.308257693",
        "0.448811996",
        "0.881147349",
        "-0.480254929",
        "0.578821984"
      ],
      [
        "0.785087462",
        "0.190879211",
        "-0.991874283",
        "0.142884708",
        "0.386870026",
        "0.277601429"
      ]
    ],
    [
      [
        "0.173860579",
        "-0.253402902",
        "0.078671473",
        "0.765914443",
        "0.945685853",
        "-0.056507260"
      ],
      [
        "0.827793890",
        "0.049908919",
        "0.099489172",
        "-0.415254967",
        "-0.551358693",
        "0.217894395"
      ]
    ],
    [
      [
        "0.033854285",
        "0.757945275",
        "0.238949304",
        "0.438245108",
        "-0.334446893",
        "0.429302173"
      ],
      [
        "0.204182853",
        "0.929988560",
        "-0.350059161",
        "-0.848899234",
        "-0.437667481",
        "0.296452200"
      ]
    ],
    [
      [
        "0.360420465",
        "0.805470901",
        "-0.149595759",
        "-0.534757414",
        "-0.455359142",
        "-0.830880569"
      ],
      [
        "0.437114881",
        "-0.119600487",
        "0.943855695",
        "0.105943474",
        "0.044285375",
        "-0.172589095"
      ]
    ],
    [
      [
        "0.554578364",
        "0.994254430",
        "-0.207607087",
        "-0.476824317",
        "0.833951004",
        "-0.910616681"
      ],
      [
        "0.130498709",
        "0.134857521",
        "0.359614099",
        "-0.819176124",
        "-0.188781866",
        "0.197245211"
      ]
    ],
    [
      [
        "0.207907733",
        "-0.071207230",
        "-0.050195652",
        "0.361311424",
        "0.629145577",
        "0.676341414"
      ],
      [
        "0.134928052",
        "0.654983616",
        "-0.164260034",
        "0.661474242",
        "-0.649906976",
        "-0.738640991"
      ]
    ],
    [
      [
        "0.380773321",
        "-0.230612485",
        "0.828823822",
        "-0.302137492",
        "-0.703337115",
        "0.116785538"
      ],
      [
        "0.164965248",
        "0.826698909",
        "0.552945624",
        "0.282394660",
        "0.402826360",
        "0.750848139"
      ]
    ],
    [
      [
        "0.630489438",
        "-0.574754713",
        "0.481340695",
        "0.308423087",
        "-0.380026497",
        "-0.284757812"
      ],
      [
        "0.857761883",
        "-0.467225664",
        "0.453747279",
        "-0.937809737",
        "0.540031263",
        "0.120052756"
      ]
    ],
    [
      [
        "0.677850118",
        "-0.942646756",
        "0.130539554",
        "0.165403325",
        "0.144472613",
        "0.866371766"
      ],
      [
        "0.390808129",
        "-0.431950745",
        "-0.363629095",
        "0.151008660",
        "0.147015335",
        "-0.976546017"
      ]
    ],
    [
      [
        "0.241987594",
        "-0.212666150",
        "-0.819999111",
        "0.371478787",
        "-0.317923899",
        "0.900352521"
      ],
      [
        "0.925001583",
        "-0.722520538",
        "-0.320205734",
        "-0.951005329",
        "-0.503615768",
        "-0.103984791"
      ]
    ],
    [
      [
        "0.799617216",
        "0.685518677",
        "-0.273510234",
        "0.188212646",
        "-0.749330893",
        "-0.153061269"
      ],
      [
        "0.514793507",
        "-0.966608151",
        "-0.103795319",
        "0.148429063",
        "-0.842036626",
        "0.192214407"
      ]
    ],
    [
      [
        "0.501700619",
        "-0.630635106",
        "-0.941520545",
        "0.562417799",
        "-0.470199155",
        "-0.320196694"
      ],
      [
        "0.234337612",
        "0.452882174",
        "0.883354908",
        "0.417236495",
        "0.441780915",
        "-0.138072359"
      ]
    ],
    [
      [
        "0.543895573",
        "-0.043651670",
        "-0.872613388",
        "-0.368633003",
        "-0.625896534",
        "-0.720562789"
      ],
      [
        "0.882390705",
        "-0.036946036",
        "-0.635323156",
        "-0.609106719",
        "0.677560181",
        "-0.695433859"
      ]
    ],
    [
      [
        "0.148506435",
        "0.114525850",
        "0.140279562",
        "-0.042276094",
        "-0.853059414",
        "0.517489010"
      ],
      [
        "0.182730986",
        "0.230320362",
        "0.621102229",
        "0.453308682",
        "-0.437048296",
        "-0.717226410"
      ]
    ],
    [
      [
        "0.049173651",
        "-0.804562703",
        "-0.651927470",
        "-0.028032235",
        "0.238543933",
        "0.273858848"
      ],
      [
        "0.881981975",
        "-0.514930328",
        "-0.686190501",
        "-0.391210656",
        "0.750021205",
        "0.611831926"
      ]
    ],
    [
      [
        "0.462942077",
        "-0.325939805",
        "0.360881381",
        "0.028648666",
        "0.243208775",
        "-0.686963356"
      ],
      [
        "0.633656189",
        "0.810905443",
        "-0.363646434",
        "0.280529830",
        "0.324692054",
        "0.254617542"
      ]
    ]
  ]
}

提交 subspaces：恰好 n 个子空间，每个是 2 个长度为 d 的基向量（十进制字符串，[-1, 1]）。基的选择不影响分数：同一个平面的任何基得同一个分。 · 验证器 v1.0.0

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