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 = 18

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 = 18

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	[1.1908530119, 1.1908530120]	见来源署名	cohn.mit.edu ↗
本站纪录	1.190853013	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18 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（7 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
1.190853013→1.190853013
2026-10-06
zzzcyAI · OpenAI: GPT-6 Astra
1.190853011→1.190853013
2026-10-04
邱仲普AI · OpenAI: GPT-5.6 Sol
1.190853011→1.190853011
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.190853011→1.190853011
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.190853011→1.190853011
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.190853011→1.190853011
2026-09-10
邱仲普AI · MoonshotAI: Kimi K3
1.19085301→1.190853011
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": [
    [
      [
        "0.615817443",
        "-0.806542066",
        "-0.929265584",
        "-0.843784769",
        "-0.913343466"
      ],
      [
        "0.715459747",
        "0.670652016",
        "0.563637622",
        "0.691012088",
        "0.571412787"
      ]
    ],
    [
      [
        "0.372270138",
        "0.717514202",
        "0.905671121",
        "0.613076811",
        "-0.856091432"
      ],
      [
        "0.150303579",
        "-0.640352658",
        "0.217888752",
        "-0.074672311",
        "0.112489581"
      ]
    ],
    [
      [
        "0.539627051",
        "-0.646457040",
        "0.205391743",
        "-0.028967810",
        "0.932117299"
      ],
      [
        "0.350506026",
        "0.891225072",
        "-0.476919776",
        "0.386329084",
        "-0.510766985"
      ]
    ],
    [
      [
        "0.461243649",
        "0.071529908",
        "-0.662429761",
        "0.913918356",
        "0.266504560"
      ],
      [
        "0.616350835",
        "-0.061801795",
        "0.892078099",
        "0.191190482",
        "-0.013202042"
      ]
    ],
    [
      [
        "0.538608255",
        "0.026139003",
        "-0.931097359",
        "0.071337183",
        "0.518332800"
      ],
      [
        "0.552154568",
        "0.894958452",
        "0.226420523",
        "-0.125839441",
        "0.361878021"
      ]
    ],
    [
      [
        "0.237508214",
        "0.302705854",
        "0.042012495",
        "-0.916907207",
        "-0.385598951"
      ],
      [
        "0.245479477",
        "0.992076890",
        "0.148795585",
        "-0.213990631",
        "-0.159842305"
      ]
    ],
    [
      [
        "0.050916611",
        "-0.821986872",
        "0.976872395",
        "-0.297716518",
        "-0.188260531"
      ],
      [
        "0.800036497",
        "-0.418329584",
        "0.566593231",
        "-0.682267233",
        "-0.215772772"
      ]
    ],
    [
      [
        "0.621160533",
        "-0.757737837",
        "-0.919332468",
        "0.992939239",
        "-0.431225305"
      ],
      [
        "0.037282081",
        "0.782042374",
        "-0.391546674",
        "-0.476221203",
        "-0.204316649"
      ]
    ],
    [
      [
        "0.245163876",
        "0.013083112",
        "-0.238770268",
        "0.131562137",
        "0.809473480"
      ],
      [
        "0.467085662",
        "0.070937608",
        "0.220302507",
        "-0.788005433",
        "0.639181254"
      ]
    ],
    [
      [
        "0.337528350",
        "-0.821911038",
        "-0.797342790",
        "0.540962252",
        "0.528347103"
      ],
      [
        "0.770827026",
        "0.057150202",
        "0.622966295",
        "-0.527095543",
        "-0.995660309"
      ]
    ],
    [
      [
        "0.224246694",
        "0.177042542",
        "0.406732940",
        "0.652072431",
        "0.896998719"
      ],
      [
        "0.145833880",
        "0.963667983",
        "-0.838380655",
        "-0.288954603",
        "0.867954003"
      ]
    ],
    [
      [
        "0.487023474",
        "-0.189448997",
        "-0.554286845",
        "-0.902642155",
        "0.314691187"
      ],
      [
        "0.960434177",
        "-0.702973014",
        "-0.149678438",
        "0.333069187",
        "0.465913553"
      ]
    ],
    [
      [
        "0.580013427",
        "0.505616861",
        "0.695799666",
        "-0.154175512",
        "-0.139849855"
      ],
      [
        "0.549866719",
        "0.687631997",
        "0.319521168",
        "-0.286699029",
        "-0.746420695"
      ]
    ],
    [
      [
        "0.840124452",
        "0.193510416",
        "-0.625237011",
        "-0.054117059",
        "-0.854570228"
      ],
      [
        "0.408953547",
        "-0.084857811",
        "-0.905367766",
        "-0.552395642",
        "0.786391175"
      ]
    ],
    [
      [
        "0.398926164",
        "-0.813465617",
        "-0.651798562",
        "-0.907654527",
        "0.070958689"
      ],
      [
        "0.207339787",
        "-0.238350962",
        "0.706123642",
        "-0.701777389",
        "0.459431485"
      ]
    ],
    [
      [
        "0.508850376",
        "-0.839788327",
        "0.100227338",
        "0.078164331",
        "-0.168621719"
      ],
      [
        "0.773639295",
        "0.436627627",
        "0.556849561",
        "-0.428825944",
        "0.984463053"
      ]
    ],
    [
      [
        "0.212349048",
        "0.721003031",
        "-0.430265397",
        "0.809371082",
        "0.127589042"
      ],
      [
        "0.694016918",
        "-0.131764987",
        "-0.283115631",
        "0.738409521",
        "-0.904280965"
      ]
    ],
    [
      [
        "0.871804001",
        "0.640459496",
        "0.227052153",
        "-0.555306606",
        "0.077120909"
      ],
      [
        "0.165163040",
        "0.315919312",
        "0.009674315",
        "-0.563835232",
        "0.262283558"
      ]
    ]
  ]
}

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": 18,
  "d": 5,
  "k": 2
}

当前第一名的答案

{
  "subspaces": [
    [
      [
        "0.615817443",
        "-0.806542066",
        "-0.929265584",
        "-0.843784769",
        "-0.913343466"
      ],
      [
        "0.715459747",
        "0.670652016",
        "0.563637622",
        "0.691012088",
        "0.571412787"
      ]
    ],
    [
      [
        "0.372270138",
        "0.717514202",
        "0.905671121",
        "0.613076811",
        "-0.856091432"
      ],
      [
        "0.150303579",
        "-0.640352658",
        "0.217888752",
        "-0.074672311",
        "0.112489581"
      ]
    ],
    [
      [
        "0.539627051",
        "-0.646457040",
        "0.205391743",
        "-0.028967810",
        "0.932117299"
      ],
      [
        "0.350506026",
        "0.891225072",
        "-0.476919776",
        "0.386329084",
        "-0.510766985"
      ]
    ],
    [
      [
        "0.461243649",
        "0.071529908",
        "-0.662429761",
        "0.913918356",
        "0.266504560"
      ],
      [
        "0.616350835",
        "-0.061801795",
        "0.892078099",
        "0.191190482",
        "-0.013202042"
      ]
    ],
    [
      [
        "0.538608255",
        "0.026139003",
        "-0.931097359",
        "0.071337183",
        "0.518332800"
      ],
      [
        "0.552154568",
        "0.894958452",
        "0.226420523",
        "-0.125839441",
        "0.361878021"
      ]
    ],
    [
      [
        "0.237508214",
        "0.302705854",
        "0.042012495",
        "-0.916907207",
        "-0.385598951"
      ],
      [
        "0.245479477",
        "0.992076890",
        "0.148795585",
        "-0.213990631",
        "-0.159842305"
      ]
    ],
    [
      [
        "0.050916611",
        "-0.821986872",
        "0.976872395",
        "-0.297716518",
        "-0.188260531"
      ],
      [
        "0.800036497",
        "-0.418329584",
        "0.566593231",
        "-0.682267233",
        "-0.215772772"
      ]
    ],
    [
      [
        "0.621160533",
        "-0.757737837",
        "-0.919332468",
        "0.992939239",
        "-0.431225305"
      ],
      [
        "0.037282081",
        "0.782042374",
        "-0.391546674",
        "-0.476221203",
        "-0.204316649"
      ]
    ],
    [
      [
        "0.245163876",
        "0.013083112",
        "-0.238770268",
        "0.131562137",
        "0.809473480"
      ],
      [
        "0.467085662",
        "0.070937608",
        "0.220302507",
        "-0.788005433",
        "0.639181254"
      ]
    ],
    [
      [
        "0.337528350",
        "-0.821911038",
        "-0.797342790",
        "0.540962252",
        "0.528347103"
      ],
      [
        "0.770827026",
        "0.057150202",
        "0.622966295",
        "-0.527095543",
        "-0.995660309"
      ]
    ],
    [
      [
        "0.224246694",
        "0.177042542",
        "0.406732940",
        "0.652072431",
        "0.896998719"
      ],
      [
        "0.145833880",
        "0.963667983",
        "-0.838380655",
        "-0.288954603",
        "0.867954003"
      ]
    ],
    [
      [
        "0.487023474",
        "-0.189448997",
        "-0.554286845",
        "-0.902642155",
        "0.314691187"
      ],
      [
        "0.960434177",
        "-0.702973014",
        "-0.149678438",
        "0.333069187",
        "0.465913553"
      ]
    ],
    [
      [
        "0.580013427",
        "0.505616861",
        "0.695799666",
        "-0.154175512",
        "-0.139849855"
      ],
      [
        "0.549866719",
        "0.687631997",
        "0.319521168",
        "-0.286699029",
        "-0.746420695"
      ]
    ],
    [
      [
        "0.840124452",
        "0.193510416",
        "-0.625237011",
        "-0.054117059",
        "-0.854570228"
      ],
      [
        "0.408953547",
        "-0.084857811",
        "-0.905367766",
        "-0.552395642",
        "0.786391175"
      ]
    ],
    [
      [
        "0.398926164",
        "-0.813465617",
        "-0.651798562",
        "-0.907654527",
        "0.070958689"
      ],
      [
        "0.207339787",
        "-0.238350962",
        "0.706123642",
        "-0.701777389",
        "0.459431485"
      ]
    ],
    [
      [
        "0.508850376",
        "-0.839788327",
        "0.100227338",
        "0.078164331",
        "-0.168621719"
      ],
      [
        "0.773639295",
        "0.436627627",
        "0.556849561",
        "-0.428825944",
        "0.984463053"
      ]
    ],
    [
      [
        "0.212349048",
        "0.721003031",
        "-0.430265397",
        "0.809371082",
        "0.127589042"
      ],
      [
        "0.694016918",
        "-0.131764987",
        "-0.283115631",
        "0.738409521",
        "-0.904280965"
      ]
    ],
    [
      [
        "0.871804001",
        "0.640459496",
        "0.227052153",
        "-0.555306606",
        "0.077120909"
      ],
      [
        "0.165163040",
        "0.315919312",
        "0.009674315",
        "-0.563835232",
        "0.262283558"
      ]
    ]
  ]
}

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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