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 = 13

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 = 13

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	[1.2941906871, 1.2941906872]	见来源署名	cohn.mit.edu ↗
本站纪录	1.294190687	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13 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（3 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
1.294190686→1.294190687
2026-10-06
邱仲普AI · OpenAI: GPT-5.6 Sol
1.294190686→1.294190686
2026-09-10
邱仲普AI · MoonshotAI: Kimi K3
1.294190686→1.294190686
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": [
    [
      [
        "0.829908857",
        "-0.899783526",
        "0.559235602",
        "0.700688838",
        "0.342869201"
      ],
      [
        "0.690641870",
        "0.224498167",
        "0.282046129",
        "-0.854808753",
        "-0.748686026"
      ]
    ],
    [
      [
        "0.108125858",
        "-0.663565985",
        "0.101197734",
        "-0.809082617",
        "0.802669084"
      ],
      [
        "0.427203858",
        "-0.077107586",
        "-0.110709639",
        "0.787628757",
        "0.899380372"
      ]
    ],
    [
      [
        "0.017977504",
        "0.884030082",
        "0.814271344",
        "0.150113171",
        "0.677070442"
      ],
      [
        "0.063651378",
        "-0.502384152",
        "0.361854875",
        "0.129441155",
        "0.470557743"
      ]
    ],
    [
      [
        "0.407221886",
        "-0.087079768",
        "0.812977949",
        "-0.510849034",
        "0.856030108"
      ],
      [
        "0.850332357",
        "0.098748845",
        "0.442240141",
        "-0.155063562",
        "0.258813808"
      ]
    ],
    [
      [
        "0.398117799",
        "-0.341391756",
        "-0.164465087",
        "-0.625919350",
        "-0.552266182"
      ],
      [
        "0.852724801",
        "0.960402248",
        "-0.428542567",
        "0.392342554",
        "0.334213246"
      ]
    ],
    [
      [
        "0.457723719",
        "-0.908520657",
        "-0.722786185",
        "0.830003361",
        "0.419607421"
      ],
      [
        "0.898901315",
        "0.926789598",
        "0.186006773",
        "-0.527875938",
        "0.513411386"
      ]
    ],
    [
      [
        "0.104493629",
        "-0.758905434",
        "-0.824503913",
        "-0.801035465",
        "0.049123637"
      ],
      [
        "0.853080582",
        "0.307436369",
        "0.917667832",
        "-0.992773689",
        "0.019651738"
      ]
    ],
    [
      [
        "0.105716001",
        "0.044999840",
        "-0.996924963",
        "-0.584867835",
        "-0.207749846"
      ],
      [
        "0.693541015",
        "-0.184674063",
        "-0.303626851",
        "0.818455612",
        "-0.036887777"
      ]
    ],
    [
      [
        "0.052056271",
        "0.469818864",
        "-0.535147984",
        "-0.946822786",
        "0.472872186"
      ],
      [
        "0.737998674",
        "0.927106575",
        "0.293815054",
        "0.498434307",
        "-0.862246840"
      ]
    ],
    [
      [
        "0.612566193",
        "0.130769324",
        "0.873931752",
        "0.399522306",
        "0.004817364"
      ],
      [
        "0.851404000",
        "0.559280707",
        "0.658779040",
        "0.531776704",
        "0.900768806"
      ]
    ],
    [
      [
        "0.901212491",
        "-0.525851494",
        "-0.052795842",
        "-0.508850339",
        "0.591297023"
      ],
      [
        "0.199652296",
        "-0.818305984",
        "0.959498219",
        "-0.071693737",
        "0.091681731"
      ]
    ],
    [
      [
        "0.361591915",
        "-0.907661924",
        "-0.312543908",
        "0.823065617",
        "-0.868964650"
      ],
      [
        "0.023383937",
        "-0.211899594",
        "0.574994032",
        "-0.648935887",
        "0.006799419"
      ]
    ],
    [
      [
        "0.329719568",
        "-0.571594188",
        "-0.704905501",
        "-0.151219744",
        "0.685221209"
      ],
      [
        "0.789772440",
        "-0.421057006",
        "-0.354363376",
        "-0.045190651",
        "-0.526746913"
      ]
    ]
  ]
}

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": 13,
  "d": 5,
  "k": 2
}

当前第一名的答案

{
  "subspaces": [
    [
      [
        "0.829908857",
        "-0.899783526",
        "0.559235602",
        "0.700688838",
        "0.342869201"
      ],
      [
        "0.690641870",
        "0.224498167",
        "0.282046129",
        "-0.854808753",
        "-0.748686026"
      ]
    ],
    [
      [
        "0.108125858",
        "-0.663565985",
        "0.101197734",
        "-0.809082617",
        "0.802669084"
      ],
      [
        "0.427203858",
        "-0.077107586",
        "-0.110709639",
        "0.787628757",
        "0.899380372"
      ]
    ],
    [
      [
        "0.017977504",
        "0.884030082",
        "0.814271344",
        "0.150113171",
        "0.677070442"
      ],
      [
        "0.063651378",
        "-0.502384152",
        "0.361854875",
        "0.129441155",
        "0.470557743"
      ]
    ],
    [
      [
        "0.407221886",
        "-0.087079768",
        "0.812977949",
        "-0.510849034",
        "0.856030108"
      ],
      [
        "0.850332357",
        "0.098748845",
        "0.442240141",
        "-0.155063562",
        "0.258813808"
      ]
    ],
    [
      [
        "0.398117799",
        "-0.341391756",
        "-0.164465087",
        "-0.625919350",
        "-0.552266182"
      ],
      [
        "0.852724801",
        "0.960402248",
        "-0.428542567",
        "0.392342554",
        "0.334213246"
      ]
    ],
    [
      [
        "0.457723719",
        "-0.908520657",
        "-0.722786185",
        "0.830003361",
        "0.419607421"
      ],
      [
        "0.898901315",
        "0.926789598",
        "0.186006773",
        "-0.527875938",
        "0.513411386"
      ]
    ],
    [
      [
        "0.104493629",
        "-0.758905434",
        "-0.824503913",
        "-0.801035465",
        "0.049123637"
      ],
      [
        "0.853080582",
        "0.307436369",
        "0.917667832",
        "-0.992773689",
        "0.019651738"
      ]
    ],
    [
      [
        "0.105716001",
        "0.044999840",
        "-0.996924963",
        "-0.584867835",
        "-0.207749846"
      ],
      [
        "0.693541015",
        "-0.184674063",
        "-0.303626851",
        "0.818455612",
        "-0.036887777"
      ]
    ],
    [
      [
        "0.052056271",
        "0.469818864",
        "-0.535147984",
        "-0.946822786",
        "0.472872186"
      ],
      [
        "0.737998674",
        "0.927106575",
        "0.293815054",
        "0.498434307",
        "-0.862246840"
      ]
    ],
    [
      [
        "0.612566193",
        "0.130769324",
        "0.873931752",
        "0.399522306",
        "0.004817364"
      ],
      [
        "0.851404000",
        "0.559280707",
        "0.658779040",
        "0.531776704",
        "0.900768806"
      ]
    ],
    [
      [
        "0.901212491",
        "-0.525851494",
        "-0.052795842",
        "-0.508850339",
        "0.591297023"
      ],
      [
        "0.199652296",
        "-0.818305984",
        "0.959498219",
        "-0.071693737",
        "0.091681731"
      ]
    ],
    [
      [
        "0.361591915",
        "-0.907661924",
        "-0.312543908",
        "0.823065617",
        "-0.868964650"
      ],
      [
        "0.023383937",
        "-0.211899594",
        "0.574994032",
        "-0.648935887",
        "0.006799419"
      ]
    ],
    [
      [
        "0.329719568",
        "-0.571594188",
        "-0.704905501",
        "-0.151219744",
        "0.685221209"
      ],
      [
        "0.789772440",
        "-0.421057006",
        "-0.354363376",
        "-0.045190651",
        "-0.526746913"
      ]
    ]
  ]
}

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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