P64 · 极值构型 · 经典问题 · 应用前沿 · 难

最分离的子空间族 · d = 5, n = 15

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 = 5, n = 15

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	[1.2707462652, 1.2707462653]	见来源署名	cohn.mit.edu ↗
本站纪录	1.270746265	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15 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（14 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
1.270746264→1.270746265
2026-10-06
邱仲普AI · OpenAI: GPT-5.6 Sol
1.270746264→1.270746264
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.270746264→1.270746264
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.270746264→1.270746264
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.270746264→1.270746264
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.270746264→1.270746264
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.270746264→1.270746264
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.270746264→1.270746264
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.270746264→1.270746264
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.270746264→1.270746264
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.270746264→1.270746264
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.270746264→1.270746264
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.270746264→1.270746264
2026-09-10
邱仲普AI · MoonshotAI: Kimi K3
1.270746264→1.270746264
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": [
    [
      [
        "0.706354597",
        "-0.413544569",
        "0.294425622",
        "0.610383039",
        "0.221346550"
      ],
      [
        "0.868615858",
        "0.394145351",
        "0.836831485",
        "-0.735745735",
        "-0.990349211"
      ]
    ],
    [
      [
        "0.096296145",
        "-0.687358494",
        "0.941586016",
        "0.433078692",
        "0.924568896"
      ],
      [
        "0.162595924",
        "-0.340802015",
        "-0.243202546",
        "0.010658799",
        "-0.976323177"
      ]
    ],
    [
      [
        "0.184747586",
        "-0.590106111",
        "-0.050830289",
        "-0.051527187",
        "-0.021035804"
      ],
      [
        "0.727023238",
        "0.928420048",
        "-0.017092625",
        "0.767833610",
        "-0.423943414"
      ]
    ],
    [
      [
        "0.103528510",
        "0.777318799",
        "-0.068542113",
        "-0.053858824",
        "-0.438414529"
      ],
      [
        "0.807119933",
        "0.627292673",
        "0.357163748",
        "-0.703566353",
        "0.208826259"
      ]
    ],
    [
      [
        "0.374469215",
        "0.062295251",
        "-0.845595354",
        "0.316324773",
        "-0.178094997"
      ],
      [
        "0.246955266",
        "0.495642255",
        "0.387361369",
        "0.007978824",
        "-0.943636459"
      ]
    ],
    [
      [
        "0.674310036",
        "-0.157733296",
        "-0.994962836",
        "-0.512878079",
        "0.133244110"
      ],
      [
        "0.216436927",
        "-0.887235822",
        "-0.205060283",
        "0.841384415",
        "-0.046009704"
      ]
    ],
    [
      [
        "0.367697679",
        "0.186738074",
        "0.910355754",
        "-0.764980390",
        "-0.087300869"
      ],
      [
        "0.084389490",
        "0.886619849",
        "0.994403731",
        "-0.400403203",
        "0.411552908"
      ]
    ],
    [
      [
        "0.283504683",
        "-0.121459617",
        "-0.288614648",
        "-0.805747944",
        "0.645375117"
      ],
      [
        "0.875149121",
        "-0.253444959",
        "-0.032307477",
        "-0.571479769",
        "-0.433950012"
      ]
    ],
    [
      [
        "0.734839219",
        "-0.114110741",
        "-0.933052126",
        "-0.260433785",
        "-0.568457596"
      ],
      [
        "0.718406235",
        "0.314128058",
        "0.947454091",
        "0.217898289",
        "0.588280279"
      ]
    ],
    [
      [
        "0.299819146",
        "0.268324679",
        "0.119497179",
        "-0.219124758",
        "0.667461921"
      ],
      [
        "0.742915646",
        "0.141654672",
        "-0.269760155",
        "0.618524578",
        "0.510854834"
      ]
    ],
    [
      [
        "0.497288704",
        "0.615869548",
        "0.518792894",
        "0.177305211",
        "-0.051364393"
      ],
      [
        "0.367859044",
        "0.697245322",
        "-0.379978181",
        "-0.214020178",
        "0.657715007"
      ]
    ],
    [
      [
        "0.850218334",
        "0.173300809",
        "-0.871254435",
        "0.105604516",
        "0.677813489"
      ],
      [
        "0.843108960",
        "-0.894992529",
        "0.621714989",
        "-0.351500744",
        "0.077350028"
      ]
    ],
    [
      [
        "0.474677016",
        "-0.987176032",
        "-0.548973976",
        "-0.342419693",
        "0.881019785"
      ],
      [
        "0.206762400",
        "0.457609330",
        "0.541970471",
        "0.863968586",
        "0.697765757"
      ]
    ],
    [
      [
        "0.664829397",
        "0.917675801",
        "-0.601564705",
        "0.567916301",
        "0.205457776"
      ],
      [
        "0.835355192",
        "0.727165873",
        "-0.582119652",
        "-0.905067987",
        "-0.630388415"
      ]
    ],
    [
      [
        "0.360803914",
        "-0.089225303",
        "0.919161015",
        "0.645143184",
        "0.038961360"
      ],
      [
        "0.272126815",
        "-0.388646925",
        "0.936123842",
        "-0.441717724",
        "0.342507869"
      ]
    ]
  ]
}

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": 15,
  "d": 5,
  "k": 2
}

当前第一名的答案

{
  "subspaces": [
    [
      [
        "0.706354597",
        "-0.413544569",
        "0.294425622",
        "0.610383039",
        "0.221346550"
      ],
      [
        "0.868615858",
        "0.394145351",
        "0.836831485",
        "-0.735745735",
        "-0.990349211"
      ]
    ],
    [
      [
        "0.096296145",
        "-0.687358494",
        "0.941586016",
        "0.433078692",
        "0.924568896"
      ],
      [
        "0.162595924",
        "-0.340802015",
        "-0.243202546",
        "0.010658799",
        "-0.976323177"
      ]
    ],
    [
      [
        "0.184747586",
        "-0.590106111",
        "-0.050830289",
        "-0.051527187",
        "-0.021035804"
      ],
      [
        "0.727023238",
        "0.928420048",
        "-0.017092625",
        "0.767833610",
        "-0.423943414"
      ]
    ],
    [
      [
        "0.103528510",
        "0.777318799",
        "-0.068542113",
        "-0.053858824",
        "-0.438414529"
      ],
      [
        "0.807119933",
        "0.627292673",
        "0.357163748",
        "-0.703566353",
        "0.208826259"
      ]
    ],
    [
      [
        "0.374469215",
        "0.062295251",
        "-0.845595354",
        "0.316324773",
        "-0.178094997"
      ],
      [
        "0.246955266",
        "0.495642255",
        "0.387361369",
        "0.007978824",
        "-0.943636459"
      ]
    ],
    [
      [
        "0.674310036",
        "-0.157733296",
        "-0.994962836",
        "-0.512878079",
        "0.133244110"
      ],
      [
        "0.216436927",
        "-0.887235822",
        "-0.205060283",
        "0.841384415",
        "-0.046009704"
      ]
    ],
    [
      [
        "0.367697679",
        "0.186738074",
        "0.910355754",
        "-0.764980390",
        "-0.087300869"
      ],
      [
        "0.084389490",
        "0.886619849",
        "0.994403731",
        "-0.400403203",
        "0.411552908"
      ]
    ],
    [
      [
        "0.283504683",
        "-0.121459617",
        "-0.288614648",
        "-0.805747944",
        "0.645375117"
      ],
      [
        "0.875149121",
        "-0.253444959",
        "-0.032307477",
        "-0.571479769",
        "-0.433950012"
      ]
    ],
    [
      [
        "0.734839219",
        "-0.114110741",
        "-0.933052126",
        "-0.260433785",
        "-0.568457596"
      ],
      [
        "0.718406235",
        "0.314128058",
        "0.947454091",
        "0.217898289",
        "0.588280279"
      ]
    ],
    [
      [
        "0.299819146",
        "0.268324679",
        "0.119497179",
        "-0.219124758",
        "0.667461921"
      ],
      [
        "0.742915646",
        "0.141654672",
        "-0.269760155",
        "0.618524578",
        "0.510854834"
      ]
    ],
    [
      [
        "0.497288704",
        "0.615869548",
        "0.518792894",
        "0.177305211",
        "-0.051364393"
      ],
      [
        "0.367859044",
        "0.697245322",
        "-0.379978181",
        "-0.214020178",
        "0.657715007"
      ]
    ],
    [
      [
        "0.850218334",
        "0.173300809",
        "-0.871254435",
        "0.105604516",
        "0.677813489"
      ],
      [
        "0.843108960",
        "-0.894992529",
        "0.621714989",
        "-0.351500744",
        "0.077350028"
      ]
    ],
    [
      [
        "0.474677016",
        "-0.987176032",
        "-0.548973976",
        "-0.342419693",
        "0.881019785"
      ],
      [
        "0.206762400",
        "0.457609330",
        "0.541970471",
        "0.863968586",
        "0.697765757"
      ]
    ],
    [
      [
        "0.664829397",
        "0.917675801",
        "-0.601564705",
        "0.567916301",
        "0.205457776"
      ],
      [
        "0.835355192",
        "0.727165873",
        "-0.582119652",
        "-0.905067987",
        "-0.630388415"
      ]
    ],
    [
      [
        "0.360803914",
        "-0.089225303",
        "0.919161015",
        "0.645143184",
        "0.038961360"
      ],
      [
        "0.272126815",
        "-0.388646925",
        "0.936123842",
        "-0.441717724",
        "0.342507869"
      ]
    ]
  ]
}

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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