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 = 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 = 6, 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	[1.4285668598, 1.428566865268]	见来源署名	cohn.mit.edu ↗
本站纪录	1.428566865	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（5 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
1.428566864→1.428566865
2026-10-06
zzzcyAI · OpenAI: GPT-6 Astra
1.428566861→1.428566864
2026-10-04
邱仲普AI · OpenAI: GPT-5.6 Sol
1.428566861→1.428566861
2026-09-10
邱仲普AI · MoonshotAI: Kimi K3
1.428566859→1.428566861
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
1.428566858→1.428566859
2026-09-01

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": [
    [
      [
        "0.124443953",
        "0.291033324",
        "0.912748239",
        "-0.154663180",
        "-0.380715146",
        "-0.026620074"
      ],
      [
        "0.075312273",
        "0.470314190",
        "-0.307824579",
        "0.079814346",
        "-0.903797663",
        "0.214942325"
      ]
    ],
    [
      [
        "0.614200487",
        "0.336002840",
        "0.037187906",
        "0.981476641",
        "-0.530382589",
        "-0.931057864"
      ],
      [
        "0.659021466",
        "0.732817239",
        "0.559371924",
        "0.128282950",
        "0.859272669",
        "-0.904433368"
      ]
    ],
    [
      [
        "0.399174464",
        "0.076601909",
        "0.510077788",
        "-0.494478881",
        "-0.800080027",
        "0.819239846"
      ],
      [
        "0.219295389",
        "0.721912523",
        "-0.353604631",
        "-0.509665181",
        "0.556603685",
        "0.325062260"
      ]
    ],
    [
      [
        "0.797963700",
        "0.963981314",
        "-0.412197588",
        "0.581611476",
        "-0.487089739",
        "0.204403388"
      ],
      [
        "0.384738454",
        "-0.599123160",
        "0.364221281",
        "-0.661340109",
        "0.425942843",
        "0.490656661"
      ]
    ],
    [
      [
        "0.051541828",
        "0.965158685",
        "0.103720251",
        "-0.056565883",
        "-0.736968296",
        "-0.792025255"
      ],
      [
        "0.392179077",
        "-0.482252526",
        "0.377142004",
        "0.059160477",
        "-0.587989544",
        "-0.912902386"
      ]
    ],
    [
      [
        "0.829533658",
        "0.457744420",
        "-0.787991237",
        "0.748893072",
        "0.737852196",
        "-0.160128756"
      ],
      [
        "0.662410563",
        "-0.180609339",
        "0.825972155",
        "-0.699546286",
        "-0.732016263",
        "-0.441061850"
      ]
    ],
    [
      [
        "0.060711139",
        "0.272079946",
        "-0.663391396",
        "-0.285817864",
        "-0.160083699",
        "-0.164995104"
      ],
      [
        "0.433951227",
        "0.938064116",
        "0.946828790",
        "0.976522861",
        "0.861010264",
        "0.142492386"
      ]
    ],
    [
      [
        "0.594863961",
        "-0.767850113",
        "-0.181615330",
        "0.600350176",
        "0.855313244",
        "-0.829891619"
      ],
      [
        "0.093242040",
        "-0.528446947",
        "-0.132504132",
        "-0.984793764",
        "-0.049406638",
        "0.027990622"
      ]
    ],
    [
      [
        "0.440296017",
        "-0.817031767",
        "-0.653313976",
        "0.150532332",
        "-0.317051836",
        "-0.543123134"
      ],
      [
        "0.440759204",
        "0.765267782",
        "0.935869047",
        "0.253750575",
        "-0.058580740",
        "0.632467508"
      ]
    ],
    [
      [
        "0.675623358",
        "-0.456428897",
        "-0.179814422",
        "0.163085454",
        "-0.005970024",
        "0.490431628"
      ],
      [
        "0.967482889",
        "-0.478184839",
        "-0.110914774",
        "0.565848212",
        "0.873684321",
        "0.634803256"
      ]
    ],
    [
      [
        "0.475781314",
        "0.213317101",
        "-0.637502409",
        "-0.252435436",
        "-0.532734492",
        "0.925106164"
      ],
      [
        "0.870963855",
        "0.225019942",
        "-0.948322754",
        "-0.402121018",
        "-0.687715908",
        "0.620483014"
      ]
    ],
    [
      [
        "0.206286627",
        "0.196838191",
        "0.641594659",
        "-0.633351455",
        "0.273075480",
        "0.465607217"
      ],
      [
        "0.011546763",
        "0.088243746",
        "-0.228546132",
        "-0.433954559",
        "-0.434157869",
        "-0.556806176"
      ]
    ],
    [
      [
        "0.310892599",
        "0.603729466",
        "0.259465976",
        "-0.388853679",
        "0.098703486",
        "-0.175281915"
      ],
      [
        "0.900821075",
        "-0.779479986",
        "-0.993777502",
        "-0.443640041",
        "0.366232781",
        "0.263422238"
      ]
    ],
    [
      [
        "0.467827455",
        "0.124608073",
        "-0.893602622",
        "0.310965157",
        "0.304009305",
        "0.921830107"
      ],
      [
        "0.117488307",
        "-0.082564109",
        "0.670010434",
        "0.746825908",
        "-0.354915723",
        "-0.264169648"
      ]
    ],
    [
      [
        "0.813548088",
        "-0.156256288",
        "0.249620026",
        "-0.784916440",
        "0.773218697",
        "-0.888827557"
      ],
      [
        "0.383540525",
        "-0.591081146",
        "0.652173006",
        "0.554802357",
        "-0.364595262",
        "0.773314056"
      ]
    ]
  ]
}

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": 6,
  "k": 2
}

当前第一名的答案

{
  "subspaces": [
    [
      [
        "0.124443953",
        "0.291033324",
        "0.912748239",
        "-0.154663180",
        "-0.380715146",
        "-0.026620074"
      ],
      [
        "0.075312273",
        "0.470314190",
        "-0.307824579",
        "0.079814346",
        "-0.903797663",
        "0.214942325"
      ]
    ],
    [
      [
        "0.614200487",
        "0.336002840",
        "0.037187906",
        "0.981476641",
        "-0.530382589",
        "-0.931057864"
      ],
      [
        "0.659021466",
        "0.732817239",
        "0.559371924",
        "0.128282950",
        "0.859272669",
        "-0.904433368"
      ]
    ],
    [
      [
        "0.399174464",
        "0.076601909",
        "0.510077788",
        "-0.494478881",
        "-0.800080027",
        "0.819239846"
      ],
      [
        "0.219295389",
        "0.721912523",
        "-0.353604631",
        "-0.509665181",
        "0.556603685",
        "0.325062260"
      ]
    ],
    [
      [
        "0.797963700",
        "0.963981314",
        "-0.412197588",
        "0.581611476",
        "-0.487089739",
        "0.204403388"
      ],
      [
        "0.384738454",
        "-0.599123160",
        "0.364221281",
        "-0.661340109",
        "0.425942843",
        "0.490656661"
      ]
    ],
    [
      [
        "0.051541828",
        "0.965158685",
        "0.103720251",
        "-0.056565883",
        "-0.736968296",
        "-0.792025255"
      ],
      [
        "0.392179077",
        "-0.482252526",
        "0.377142004",
        "0.059160477",
        "-0.587989544",
        "-0.912902386"
      ]
    ],
    [
      [
        "0.829533658",
        "0.457744420",
        "-0.787991237",
        "0.748893072",
        "0.737852196",
        "-0.160128756"
      ],
      [
        "0.662410563",
        "-0.180609339",
        "0.825972155",
        "-0.699546286",
        "-0.732016263",
        "-0.441061850"
      ]
    ],
    [
      [
        "0.060711139",
        "0.272079946",
        "-0.663391396",
        "-0.285817864",
        "-0.160083699",
        "-0.164995104"
      ],
      [
        "0.433951227",
        "0.938064116",
        "0.946828790",
        "0.976522861",
        "0.861010264",
        "0.142492386"
      ]
    ],
    [
      [
        "0.594863961",
        "-0.767850113",
        "-0.181615330",
        "0.600350176",
        "0.855313244",
        "-0.829891619"
      ],
      [
        "0.093242040",
        "-0.528446947",
        "-0.132504132",
        "-0.984793764",
        "-0.049406638",
        "0.027990622"
      ]
    ],
    [
      [
        "0.440296017",
        "-0.817031767",
        "-0.653313976",
        "0.150532332",
        "-0.317051836",
        "-0.543123134"
      ],
      [
        "0.440759204",
        "0.765267782",
        "0.935869047",
        "0.253750575",
        "-0.058580740",
        "0.632467508"
      ]
    ],
    [
      [
        "0.675623358",
        "-0.456428897",
        "-0.179814422",
        "0.163085454",
        "-0.005970024",
        "0.490431628"
      ],
      [
        "0.967482889",
        "-0.478184839",
        "-0.110914774",
        "0.565848212",
        "0.873684321",
        "0.634803256"
      ]
    ],
    [
      [
        "0.475781314",
        "0.213317101",
        "-0.637502409",
        "-0.252435436",
        "-0.532734492",
        "0.925106164"
      ],
      [
        "0.870963855",
        "0.225019942",
        "-0.948322754",
        "-0.402121018",
        "-0.687715908",
        "0.620483014"
      ]
    ],
    [
      [
        "0.206286627",
        "0.196838191",
        "0.641594659",
        "-0.633351455",
        "0.273075480",
        "0.465607217"
      ],
      [
        "0.011546763",
        "0.088243746",
        "-0.228546132",
        "-0.433954559",
        "-0.434157869",
        "-0.556806176"
      ]
    ],
    [
      [
        "0.310892599",
        "0.603729466",
        "0.259465976",
        "-0.388853679",
        "0.098703486",
        "-0.175281915"
      ],
      [
        "0.900821075",
        "-0.779479986",
        "-0.993777502",
        "-0.443640041",
        "0.366232781",
        "0.263422238"
      ]
    ],
    [
      [
        "0.467827455",
        "0.124608073",
        "-0.893602622",
        "0.310965157",
        "0.304009305",
        "0.921830107"
      ],
      [
        "0.117488307",
        "-0.082564109",
        "0.670010434",
        "0.746825908",
        "-0.354915723",
        "-0.264169648"
      ]
    ],
    [
      [
        "0.813548088",
        "-0.156256288",
        "0.249620026",
        "-0.784916440",
        "0.773218697",
        "-0.888827557"
      ],
      [
        "0.383540525",
        "-0.591081146",
        "0.652173006",
        "0.554802357",
        "-0.364595262",
        "0.773314056"
      ]
    ]
  ]
}

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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