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 = 20

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 = 20

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	[1.3887598649, 1.3887598650]	见来源署名	cohn.mit.edu ↗
本站纪录	1.388759865	zzzcy	纪录详情 ↓

在公布精度内匹配已知最好；不代表精确相等或已证明最优。

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20 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（6 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
1.388759864→1.388759865
2026-10-06
邱仲普AI · OpenAI: GPT-5.6 Sol
1.388759864→1.388759864
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.388759864→1.388759864
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.388759864→1.388759864
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.388759864→1.388759864
2026-09-10
邱仲普AI · MoonshotAI: Kimi K3
1.388759864→1.388759864
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": [
    [
      [
        "0.371530042",
        "-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"
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    ],
    [
      [
        "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",
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        "-0.113949975",
        "-0.530891964",
        "0.819468101"
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        "0.263198604",
        "-0.117403488",
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    [
      [
        "0.357432441",
        "-0.329029477",
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        "-0.547660927",
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        "-0.916842028"
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    [
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        "0.751287694",
        "0.740509763",
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        "0.492390617",
        "0.043530539",
        "-0.834840961"
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        "0.294020770",
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    [
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        "0.999629813",
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        "0.898770372",
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        "0.575086718"
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    [
      [
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        "0.375868828",
        "-0.911138547",
        "-0.383267227",
        "-0.274641946",
        "0.635338043"
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      [
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    [
      [
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        "0.874189917",
        "0.470910540",
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        "0.962415159",
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      [
        "0.735832753",
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        "0.349904936",
        "-0.713266586",
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    ],
    [
      [
        "0.466044436",
        "-0.864298567",
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        "-0.057390757",
        "-0.303448186",
        "0.942187380"
      ]
    ]
  ]
}

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

当前第一名的答案

{
  "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",
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        "-0.911378380",
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      ],
      [
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    ],
    [
      [
        "0.460576341",
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      [
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        "0.890814567",
        "-0.651881471"
      ]
    ],
    [
      [
        "0.131457384",
        "0.714935570",
        "0.996105918",
        "-0.656171901",
        "-0.558960544",
        "-0.139372049"
      ],
      [
        "0.358129202",
        "0.836581519",
        "0.760183170",
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        "-0.987025985"
      ]
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    [
      [
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        "-0.339001161",
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        "0.304821167",
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      ],
      [
        "0.266531284",
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        "-0.007666597"
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    ],
    [
      [
        "0.945695734",
        "0.463852382",
        "0.983699127",
        "-0.918476798",
        "0.968391154",
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      [
        "0.393045310",
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        "-0.030429519",
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    ],
    [
      [
        "0.338938375",
        "-0.373365525",
        "0.797829799",
        "-0.852457513",
        "-0.281585599",
        "-0.478921646"
      ],
      [
        "0.048315326",
        "0.690985887",
        "-0.909960122",
        "0.101139366",
        "-0.955989679",
        "0.382916272"
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    [
      [
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        "-0.530891964",
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        "-0.202773401",
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    [
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        "-0.122286569",
        "0.658573632"
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    [
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    [
      [
        "0.751287694",
        "0.740509763",
        "0.367544962",
        "0.492390617",
        "0.043530539",
        "-0.834840961"
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      [
        "0.489804811",
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        "0.294020770",
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    [
      [
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        "0.999629813",
        "-0.045149546"
      ],
      [
        "0.619265322",
        "-0.156180329",
        "0.783460634",
        "0.898770372",
        "-0.917498082",
        "0.575086718"
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    ],
    [
      [
        "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"
      ]
    ]
  ]
}

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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