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 = 19

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 = 19

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	[1.1761278309, 1.1761278310]	见来源署名	cohn.mit.edu ↗
本站纪录	1.176129063	zzzcy	纪录详情 ↓

数值超出外部纪录区间 · 待审核。

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19 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（5 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
1.176129063→1.176129063
2026-10-06
zzzcyAI · OpenAI: GPT-6 Astra
1.176128991→1.176129063
2026-10-04
邱仲普AI · OpenAI: GPT-5.6 Sol
1.17612899→1.176128991
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.17612899→1.17612899
2026-09-08
邱仲普AI · MoonshotAI: Kimi K3
1.17612783→1.17612899
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": [
    [
      [
        "0.117694062",
        "-0.179395871",
        "-0.490498331",
        "0.459573273",
        "0.663956847"
      ],
      [
        "0.847338730",
        "0.126196631",
        "0.911429568",
        "-0.457536675",
        "0.683828374"
      ]
    ],
    [
      [
        "0.097871194",
        "0.475294163",
        "-0.201967055",
        "0.495948572",
        "-0.941127222"
      ],
      [
        "0.212908484",
        "-0.122381469",
        "0.607973226",
        "0.250896355",
        "-0.674905414"
      ]
    ],
    [
      [
        "0.358591025",
        "0.250520642",
        "-0.757099425",
        "-0.870722134",
        "-0.923673764"
      ],
      [
        "0.168880589",
        "0.463920024",
        "-0.458430990",
        "-0.252380133",
        "-0.080791199"
      ]
    ],
    [
      [
        "0.141038418",
        "0.347719174",
        "-0.131268387",
        "0.914205046",
        "0.568995162"
      ],
      [
        "0.748813158",
        "0.238607205",
        "0.445885177",
        "0.347893504",
        "-0.542272376"
      ]
    ],
    [
      [
        "0.163426319",
        "-0.974750379",
        "-0.154335820",
        "0.836005313",
        "0.598185529"
      ],
      [
        "0.309563571",
        "-0.538744880",
        "0.195720457",
        "-0.905872992",
        "-0.068035112"
      ]
    ],
    [
      [
        "0.631912671",
        "-0.714609968",
        "0.652960103",
        "-0.718323935",
        "-0.320833909"
      ],
      [
        "0.373319886",
        "0.396272282",
        "-0.897532516",
        "0.538186260",
        "0.052602897"
      ]
    ],
    [
      [
        "0.890079647",
        "-0.555475434",
        "0.445773215",
        "0.628685933",
        "-0.275210977"
      ],
      [
        "0.529469002",
        "-0.087940468",
        "-0.346531518",
        "-0.930712151",
        "0.519756437"
      ]
    ],
    [
      [
        "0.545038205",
        "-0.686725518",
        "0.308761093",
        "0.966157773",
        "0.934103867"
      ],
      [
        "0.380290882",
        "-0.130075451",
        "-0.269879066",
        "0.360483784",
        "-0.722475748"
      ]
    ],
    [
      [
        "0.085986463",
        "-0.965500734",
        "0.385574243",
        "-0.052738170",
        "0.706572338"
      ],
      [
        "0.944444085",
        "-0.692261725",
        "0.408119021",
        "0.022330801",
        "0.805933398"
      ]
    ],
    [
      [
        "0.786663983",
        "-0.680595415",
        "-0.968169386",
        "-0.475265029",
        "0.951833570"
      ],
      [
        "0.978962169",
        "0.139513272",
        "-0.429937966",
        "-0.658252038",
        "-0.778425720"
      ]
    ],
    [
      [
        "0.843431337",
        "0.989953870",
        "0.452540083",
        "-0.554607547",
        "0.921737766"
      ],
      [
        "0.957524952",
        "-0.110932894",
        "0.027331623",
        "-0.160534873",
        "-0.674176959"
      ]
    ],
    [
      [
        "0.117077513",
        "-0.534857354",
        "-0.185701589",
        "-0.545934046",
        "0.291794247"
      ],
      [
        "0.452478921",
        "0.906950666",
        "0.144900991",
        "0.378326516",
        "0.846147448"
      ]
    ],
    [
      [
        "0.206501304",
        "-0.104203237",
        "0.918241465",
        "0.488492357",
        "-0.213051090"
      ],
      [
        "0.158245841",
        "0.279379876",
        "0.130848818",
        "-0.250504432",
        "-0.110093308"
      ]
    ],
    [
      [
        "0.850364632",
        "-0.989550121",
        "0.665131672",
        "-0.056169512",
        "0.016816054"
      ],
      [
        "0.338303065",
        "0.518124289",
        "0.965975034",
        "0.184545027",
        "0.380133904"
      ]
    ],
    [
      [
        "0.607629221",
        "0.243111916",
        "0.801512640",
        "0.811057263",
        "0.439194653"
      ],
      [
        "0.461827578",
        "-0.036266731",
        "-0.837878654",
        "-0.176879709",
        "0.143288785"
      ]
    ],
    [
      [
        "0.802498683",
        "0.559806956",
        "-0.498471888",
        "0.750981313",
        "0.066066607"
      ],
      [
        "0.386902570",
        "0.521322242",
        "-0.156315762",
        "-0.386046431",
        "0.227077681"
      ]
    ],
    [
      [
        "0.004898149",
        "-0.778396241",
        "-0.978801340",
        "0.123671438",
        "-0.010580078"
      ],
      [
        "0.825858070",
        "0.679539915",
        "-0.071249173",
        "0.533042977",
        "-0.670224791"
      ]
    ],
    [
      [
        "0.170633612",
        "0.083627926",
        "0.624734741",
        "-0.640408202",
        "0.522458184"
      ],
      [
        "0.337530497",
        "-0.231722155",
        "-0.939820454",
        "-0.579886681",
        "-0.554524888"
      ]
    ],
    [
      [
        "0.270915918",
        "0.075668411",
        "-0.661067718",
        "0.018002735",
        "0.615255994"
      ],
      [
        "0.625593897",
        "-0.783340300",
        "-0.880142012",
        "0.399403809",
        "0.225658512"
      ]
    ]
  ]
}

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": 19,
  "d": 5,
  "k": 2
}

当前第一名的答案

{
  "subspaces": [
    [
      [
        "0.117694062",
        "-0.179395871",
        "-0.490498331",
        "0.459573273",
        "0.663956847"
      ],
      [
        "0.847338730",
        "0.126196631",
        "0.911429568",
        "-0.457536675",
        "0.683828374"
      ]
    ],
    [
      [
        "0.097871194",
        "0.475294163",
        "-0.201967055",
        "0.495948572",
        "-0.941127222"
      ],
      [
        "0.212908484",
        "-0.122381469",
        "0.607973226",
        "0.250896355",
        "-0.674905414"
      ]
    ],
    [
      [
        "0.358591025",
        "0.250520642",
        "-0.757099425",
        "-0.870722134",
        "-0.923673764"
      ],
      [
        "0.168880589",
        "0.463920024",
        "-0.458430990",
        "-0.252380133",
        "-0.080791199"
      ]
    ],
    [
      [
        "0.141038418",
        "0.347719174",
        "-0.131268387",
        "0.914205046",
        "0.568995162"
      ],
      [
        "0.748813158",
        "0.238607205",
        "0.445885177",
        "0.347893504",
        "-0.542272376"
      ]
    ],
    [
      [
        "0.163426319",
        "-0.974750379",
        "-0.154335820",
        "0.836005313",
        "0.598185529"
      ],
      [
        "0.309563571",
        "-0.538744880",
        "0.195720457",
        "-0.905872992",
        "-0.068035112"
      ]
    ],
    [
      [
        "0.631912671",
        "-0.714609968",
        "0.652960103",
        "-0.718323935",
        "-0.320833909"
      ],
      [
        "0.373319886",
        "0.396272282",
        "-0.897532516",
        "0.538186260",
        "0.052602897"
      ]
    ],
    [
      [
        "0.890079647",
        "-0.555475434",
        "0.445773215",
        "0.628685933",
        "-0.275210977"
      ],
      [
        "0.529469002",
        "-0.087940468",
        "-0.346531518",
        "-0.930712151",
        "0.519756437"
      ]
    ],
    [
      [
        "0.545038205",
        "-0.686725518",
        "0.308761093",
        "0.966157773",
        "0.934103867"
      ],
      [
        "0.380290882",
        "-0.130075451",
        "-0.269879066",
        "0.360483784",
        "-0.722475748"
      ]
    ],
    [
      [
        "0.085986463",
        "-0.965500734",
        "0.385574243",
        "-0.052738170",
        "0.706572338"
      ],
      [
        "0.944444085",
        "-0.692261725",
        "0.408119021",
        "0.022330801",
        "0.805933398"
      ]
    ],
    [
      [
        "0.786663983",
        "-0.680595415",
        "-0.968169386",
        "-0.475265029",
        "0.951833570"
      ],
      [
        "0.978962169",
        "0.139513272",
        "-0.429937966",
        "-0.658252038",
        "-0.778425720"
      ]
    ],
    [
      [
        "0.843431337",
        "0.989953870",
        "0.452540083",
        "-0.554607547",
        "0.921737766"
      ],
      [
        "0.957524952",
        "-0.110932894",
        "0.027331623",
        "-0.160534873",
        "-0.674176959"
      ]
    ],
    [
      [
        "0.117077513",
        "-0.534857354",
        "-0.185701589",
        "-0.545934046",
        "0.291794247"
      ],
      [
        "0.452478921",
        "0.906950666",
        "0.144900991",
        "0.378326516",
        "0.846147448"
      ]
    ],
    [
      [
        "0.206501304",
        "-0.104203237",
        "0.918241465",
        "0.488492357",
        "-0.213051090"
      ],
      [
        "0.158245841",
        "0.279379876",
        "0.130848818",
        "-0.250504432",
        "-0.110093308"
      ]
    ],
    [
      [
        "0.850364632",
        "-0.989550121",
        "0.665131672",
        "-0.056169512",
        "0.016816054"
      ],
      [
        "0.338303065",
        "0.518124289",
        "0.965975034",
        "0.184545027",
        "0.380133904"
      ]
    ],
    [
      [
        "0.607629221",
        "0.243111916",
        "0.801512640",
        "0.811057263",
        "0.439194653"
      ],
      [
        "0.461827578",
        "-0.036266731",
        "-0.837878654",
        "-0.176879709",
        "0.143288785"
      ]
    ],
    [
      [
        "0.802498683",
        "0.559806956",
        "-0.498471888",
        "0.750981313",
        "0.066066607"
      ],
      [
        "0.386902570",
        "0.521322242",
        "-0.156315762",
        "-0.386046431",
        "0.227077681"
      ]
    ],
    [
      [
        "0.004898149",
        "-0.778396241",
        "-0.978801340",
        "0.123671438",
        "-0.010580078"
      ],
      [
        "0.825858070",
        "0.679539915",
        "-0.071249173",
        "0.533042977",
        "-0.670224791"
      ]
    ],
    [
      [
        "0.170633612",
        "0.083627926",
        "0.624734741",
        "-0.640408202",
        "0.522458184"
      ],
      [
        "0.337530497",
        "-0.231722155",
        "-0.939820454",
        "-0.579886681",
        "-0.554524888"
      ]
    ],
    [
      [
        "0.270915918",
        "0.075668411",
        "-0.661067718",
        "0.018002735",
        "0.615255994"
      ],
      [
        "0.625593897",
        "-0.783340300",
        "-0.880142012",
        "0.399403809",
        "0.225658512"
      ]
    ]
  ]
}

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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