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 = 14

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 = 14

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	[1.2790114729, 1.2790114730]	见来源署名	cohn.mit.edu ↗
本站纪录	1.279011472	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14 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（2 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
1.279011472→1.279011472
2026-10-06
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
1.279011471→1.279011472
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": [
    [
      [
        "0.117286373",
        "0.805912412",
        "-0.961481575",
        "-0.327095709",
        "0.048014863"
      ],
      [
        "0.990155656",
        "-0.763594890",
        "0.759253877",
        "0.219553905",
        "0.671144495"
      ]
    ],
    [
      [
        "0.138977908",
        "-0.907889704",
        "0.713384734",
        "-0.356823372",
        "-0.531904825"
      ],
      [
        "0.204441935",
        "-0.968732863",
        "-0.953390230",
        "-0.408963959",
        "-0.368846884"
      ]
    ],
    [
      [
        "0.606729818",
        "0.874466128",
        "0.704246920",
        "-0.771850065",
        "0.295167685"
      ],
      [
        "0.954137276",
        "-0.982976119",
        "-0.837425274",
        "0.517914338",
        "-0.297595334"
      ]
    ],
    [
      [
        "0.884145072",
        "-0.439597187",
        "0.764510077",
        "-0.479536856",
        "0.067472712"
      ],
      [
        "0.175330638",
        "0.725465098",
        "-0.182945931",
        "0.581695613",
        "-0.333600401"
      ]
    ],
    [
      [
        "0.956694688",
        "0.786163394",
        "0.479436865",
        "-0.618650113",
        "-0.800965411"
      ],
      [
        "0.001557878",
        "-0.051581704",
        "-0.958789014",
        "-0.626027189",
        "0.602099586"
      ]
    ],
    [
      [
        "0.044183978",
        "0.288953489",
        "-0.114796328",
        "-0.565956807",
        "-0.554222117"
      ],
      [
        "0.116855378",
        "-0.260723406",
        "0.751917441",
        "-0.128217596",
        "0.509750641"
      ]
    ],
    [
      [
        "0.376672375",
        "-0.020168579",
        "-0.756405760",
        "0.833419800",
        "0.899308093"
      ],
      [
        "0.857128203",
        "-0.422893604",
        "0.788459453",
        "0.133930363",
        "-0.991535282"
      ]
    ],
    [
      [
        "0.633012360",
        "-0.404260847",
        "-0.928101388",
        "-0.096692966",
        "0.571242245"
      ],
      [
        "0.556105925",
        "0.276489322",
        "-0.778829704",
        "0.693537095",
        "-0.741659307"
      ]
    ],
    [
      [
        "0.248277141",
        "0.237644420",
        "0.142622773",
        "0.517450049",
        "0.235660988"
      ],
      [
        "0.941402203",
        "-0.009435000",
        "-0.961488486",
        "-0.824328045",
        "-0.768203217"
      ]
    ],
    [
      [
        "0.071907308",
        "0.221992694",
        "0.297077608",
        "-0.218860466",
        "0.474444717"
      ],
      [
        "0.571819497",
        "0.038771670",
        "0.725601188",
        "0.888640718",
        "-0.210473120"
      ]
    ],
    [
      [
        "0.776081578",
        "-0.706306557",
        "0.152338494",
        "-0.912299210",
        "-0.984546814"
      ],
      [
        "0.519276408",
        "-0.070620696",
        "0.005059119",
        "-0.535222942",
        "0.653295100"
      ]
    ],
    [
      [
        "0.054926610",
        "-0.835936710",
        "-0.394314669",
        "0.248606608",
        "0.835245826"
      ],
      [
        "0.593573583",
        "0.870951594",
        "0.385347572",
        "0.232613804",
        "0.439467103"
      ]
    ],
    [
      [
        "0.485313775",
        "-0.886028408",
        "0.039980938",
        "0.691163997",
        "0.222003203"
      ],
      [
        "0.395238743",
        "-0.813438312",
        "-0.193417718",
        "-0.444830734",
        "0.225703758"
      ]
    ],
    [
      [
        "0.332729311",
        "0.555564410",
        "-0.294600182",
        "-0.073670782",
        "-0.554938445"
      ],
      [
        "0.410388963",
        "-0.988560818",
        "0.110450062",
        "0.335023302",
        "-0.638028494"
      ]
    ]
  ]
}

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": 14,
  "d": 5,
  "k": 2
}

当前第一名的答案

{
  "subspaces": [
    [
      [
        "0.117286373",
        "0.805912412",
        "-0.961481575",
        "-0.327095709",
        "0.048014863"
      ],
      [
        "0.990155656",
        "-0.763594890",
        "0.759253877",
        "0.219553905",
        "0.671144495"
      ]
    ],
    [
      [
        "0.138977908",
        "-0.907889704",
        "0.713384734",
        "-0.356823372",
        "-0.531904825"
      ],
      [
        "0.204441935",
        "-0.968732863",
        "-0.953390230",
        "-0.408963959",
        "-0.368846884"
      ]
    ],
    [
      [
        "0.606729818",
        "0.874466128",
        "0.704246920",
        "-0.771850065",
        "0.295167685"
      ],
      [
        "0.954137276",
        "-0.982976119",
        "-0.837425274",
        "0.517914338",
        "-0.297595334"
      ]
    ],
    [
      [
        "0.884145072",
        "-0.439597187",
        "0.764510077",
        "-0.479536856",
        "0.067472712"
      ],
      [
        "0.175330638",
        "0.725465098",
        "-0.182945931",
        "0.581695613",
        "-0.333600401"
      ]
    ],
    [
      [
        "0.956694688",
        "0.786163394",
        "0.479436865",
        "-0.618650113",
        "-0.800965411"
      ],
      [
        "0.001557878",
        "-0.051581704",
        "-0.958789014",
        "-0.626027189",
        "0.602099586"
      ]
    ],
    [
      [
        "0.044183978",
        "0.288953489",
        "-0.114796328",
        "-0.565956807",
        "-0.554222117"
      ],
      [
        "0.116855378",
        "-0.260723406",
        "0.751917441",
        "-0.128217596",
        "0.509750641"
      ]
    ],
    [
      [
        "0.376672375",
        "-0.020168579",
        "-0.756405760",
        "0.833419800",
        "0.899308093"
      ],
      [
        "0.857128203",
        "-0.422893604",
        "0.788459453",
        "0.133930363",
        "-0.991535282"
      ]
    ],
    [
      [
        "0.633012360",
        "-0.404260847",
        "-0.928101388",
        "-0.096692966",
        "0.571242245"
      ],
      [
        "0.556105925",
        "0.276489322",
        "-0.778829704",
        "0.693537095",
        "-0.741659307"
      ]
    ],
    [
      [
        "0.248277141",
        "0.237644420",
        "0.142622773",
        "0.517450049",
        "0.235660988"
      ],
      [
        "0.941402203",
        "-0.009435000",
        "-0.961488486",
        "-0.824328045",
        "-0.768203217"
      ]
    ],
    [
      [
        "0.071907308",
        "0.221992694",
        "0.297077608",
        "-0.218860466",
        "0.474444717"
      ],
      [
        "0.571819497",
        "0.038771670",
        "0.725601188",
        "0.888640718",
        "-0.210473120"
      ]
    ],
    [
      [
        "0.776081578",
        "-0.706306557",
        "0.152338494",
        "-0.912299210",
        "-0.984546814"
      ],
      [
        "0.519276408",
        "-0.070620696",
        "0.005059119",
        "-0.535222942",
        "0.653295100"
      ]
    ],
    [
      [
        "0.054926610",
        "-0.835936710",
        "-0.394314669",
        "0.248606608",
        "0.835245826"
      ],
      [
        "0.593573583",
        "0.870951594",
        "0.385347572",
        "0.232613804",
        "0.439467103"
      ]
    ],
    [
      [
        "0.485313775",
        "-0.886028408",
        "0.039980938",
        "0.691163997",
        "0.222003203"
      ],
      [
        "0.395238743",
        "-0.813438312",
        "-0.193417718",
        "-0.444830734",
        "0.225703758"
      ]
    ],
    [
      [
        "0.332729311",
        "0.555564410",
        "-0.294600182",
        "-0.073670782",
        "-0.554938445"
      ],
      [
        "0.410388963",
        "-0.988560818",
        "0.110450062",
        "0.335023302",
        "-0.638028494"
      ]
    ]
  ]
}

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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