P64 · 极值构型 · 经典问题 · 应用前沿 · 难

最分离的子空间族 · d = 4, 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 = 4, 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	[0.9090909090, 0.9090909091]	见来源署名	cohn.mit.edu ↗
本站纪录	0.909090909	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（2 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
0.909090908→0.909090909
2026-10-06
邱仲普AI · MoonshotAI: Kimi K3
0.909090908→0.909090908
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": [
    [
      [
        "0.281905641",
        "-0.989279720",
        "-0.068227335",
        "0.213818755"
      ],
      [
        "0.658064023",
        "0.753655396",
        "-0.988054132",
        "-0.946372402"
      ]
    ],
    [
      [
        "0.299665757",
        "0.172292481",
        "0.488567352",
        "0.671500007"
      ],
      [
        "0.856494679",
        "-0.465806475",
        "-0.458192578",
        "-0.364610798"
      ]
    ],
    [
      [
        "0.484497245",
        "0.411370832",
        "0.766488854",
        "-0.426105698"
      ],
      [
        "0.758031427",
        "-0.296435971",
        "-0.545643018",
        "-0.765765359"
      ]
    ],
    [
      [
        "0.391297183",
        "-0.833975744",
        "-0.021334392",
        "-0.147333772"
      ],
      [
        "0.028417828",
        "-0.709206661",
        "-0.576424285",
        "0.944507935"
      ]
    ],
    [
      [
        "0.702875191",
        "-0.728405378",
        "-0.785228474",
        "0.770864002"
      ],
      [
        "0.872244017",
        "-0.111201920",
        "0.613548560",
        "0.109890137"
      ]
    ],
    [
      [
        "0.227618407",
        "0.006519835",
        "-0.422602804",
        "0.962582542"
      ],
      [
        "0.889690007",
        "0.058159590",
        "-0.092847388",
        "-0.296901155"
      ]
    ],
    [
      [
        "0.824000536",
        "-0.974263551",
        "0.573647562",
        "0.022756677"
      ],
      [
        "0.315186843",
        "0.651553837",
        "-0.415016740",
        "0.712957449"
      ]
    ],
    [
      [
        "0.512885194",
        "-0.199863336",
        "0.221106399",
        "0.332853306"
      ],
      [
        "0.693301616",
        "0.945326113",
        "-0.246889754",
        "-0.659196540"
      ]
    ],
    [
      [
        "0.468667929",
        "0.699032782",
        "0.998703168",
        "-0.755960577"
      ],
      [
        "0.059680845",
        "0.453434319",
        "-0.439941727",
        "-0.157518144"
      ]
    ],
    [
      [
        "0.317588859",
        "0.871251264",
        "0.897115084",
        "0.386942132"
      ],
      [
        "0.058937777",
        "-0.256820785",
        "-0.002575911",
        "0.395875496"
      ]
    ],
    [
      [
        "0.995213684",
        "0.322752770",
        "0.029650489",
        "-0.215906106"
      ],
      [
        "0.232619277",
        "-0.929646272",
        "-0.040669006",
        "-0.458542083"
      ]
    ],
    [
      [
        "0.040849614",
        "-0.839012852",
        "-0.603944064",
        "-0.372943887"
      ],
      [
        "0.278392886",
        "0.235091616",
        "0.869197711",
        "0.240748337"
      ]
    ],
    [
      [
        "0.609097846",
        "-0.051589405",
        "-0.941615074",
        "0.127186036"
      ],
      [
        "0.568489212",
        "-0.881278531",
        "-0.606003249",
        "-0.792346023"
      ]
    ],
    [
      [
        "0.276747634",
        "0.428063205",
        "0.756431312",
        "0.433926607"
      ],
      [
        "0.779824716",
        "0.623727160",
        "-0.878782257",
        "-0.255830614"
      ]
    ],
    [
      [
        "0.407178730",
        "0.330219151",
        "-0.884189622",
        "-0.106293881"
      ],
      [
        "0.873656041",
        "-0.527970734",
        "0.785813330",
        "-0.195353294"
      ]
    ],
    [
      [
        "0.907090387",
        "0.681590297",
        "-0.468562337",
        "0.432082767"
      ],
      [
        "0.731798253",
        "-0.780664533",
        "-0.866637204",
        "0.828314965"
      ]
    ],
    [
      [
        "0.199420267",
        "-0.484781897",
        "0.764885700",
        "0.975298565"
      ],
      [
        "0.538837199",
        "0.128198194",
        "-0.932356074",
        "0.235950002"
      ]
    ],
    [
      [
        "0.815431234",
        "0.728340932",
        "0.937117801",
        "-0.064333123"
      ],
      [
        "0.295666797",
        "0.590914743",
        "-0.234049318",
        "0.790939790"
      ]
    ],
    [
      [
        "0.407400743",
        "-0.260580431",
        "0.665125349",
        "0.644797139"
      ],
      [
        "0.693948242",
        "-0.238226633",
        "0.661739590",
        "-0.793224876"
      ]
    ],
    [
      [
        "0.341934839",
        "0.746595756",
        "-0.347521054",
        "-0.298860666"
      ],
      [
        "0.248922129",
        "0.764502617",
        "-0.271035704",
        "0.755327549"
      ]
    ]
  ]
}

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": 4,
  "k": 2
}

当前第一名的答案

{
  "subspaces": [
    [
      [
        "0.281905641",
        "-0.989279720",
        "-0.068227335",
        "0.213818755"
      ],
      [
        "0.658064023",
        "0.753655396",
        "-0.988054132",
        "-0.946372402"
      ]
    ],
    [
      [
        "0.299665757",
        "0.172292481",
        "0.488567352",
        "0.671500007"
      ],
      [
        "0.856494679",
        "-0.465806475",
        "-0.458192578",
        "-0.364610798"
      ]
    ],
    [
      [
        "0.484497245",
        "0.411370832",
        "0.766488854",
        "-0.426105698"
      ],
      [
        "0.758031427",
        "-0.296435971",
        "-0.545643018",
        "-0.765765359"
      ]
    ],
    [
      [
        "0.391297183",
        "-0.833975744",
        "-0.021334392",
        "-0.147333772"
      ],
      [
        "0.028417828",
        "-0.709206661",
        "-0.576424285",
        "0.944507935"
      ]
    ],
    [
      [
        "0.702875191",
        "-0.728405378",
        "-0.785228474",
        "0.770864002"
      ],
      [
        "0.872244017",
        "-0.111201920",
        "0.613548560",
        "0.109890137"
      ]
    ],
    [
      [
        "0.227618407",
        "0.006519835",
        "-0.422602804",
        "0.962582542"
      ],
      [
        "0.889690007",
        "0.058159590",
        "-0.092847388",
        "-0.296901155"
      ]
    ],
    [
      [
        "0.824000536",
        "-0.974263551",
        "0.573647562",
        "0.022756677"
      ],
      [
        "0.315186843",
        "0.651553837",
        "-0.415016740",
        "0.712957449"
      ]
    ],
    [
      [
        "0.512885194",
        "-0.199863336",
        "0.221106399",
        "0.332853306"
      ],
      [
        "0.693301616",
        "0.945326113",
        "-0.246889754",
        "-0.659196540"
      ]
    ],
    [
      [
        "0.468667929",
        "0.699032782",
        "0.998703168",
        "-0.755960577"
      ],
      [
        "0.059680845",
        "0.453434319",
        "-0.439941727",
        "-0.157518144"
      ]
    ],
    [
      [
        "0.317588859",
        "0.871251264",
        "0.897115084",
        "0.386942132"
      ],
      [
        "0.058937777",
        "-0.256820785",
        "-0.002575911",
        "0.395875496"
      ]
    ],
    [
      [
        "0.995213684",
        "0.322752770",
        "0.029650489",
        "-0.215906106"
      ],
      [
        "0.232619277",
        "-0.929646272",
        "-0.040669006",
        "-0.458542083"
      ]
    ],
    [
      [
        "0.040849614",
        "-0.839012852",
        "-0.603944064",
        "-0.372943887"
      ],
      [
        "0.278392886",
        "0.235091616",
        "0.869197711",
        "0.240748337"
      ]
    ],
    [
      [
        "0.609097846",
        "-0.051589405",
        "-0.941615074",
        "0.127186036"
      ],
      [
        "0.568489212",
        "-0.881278531",
        "-0.606003249",
        "-0.792346023"
      ]
    ],
    [
      [
        "0.276747634",
        "0.428063205",
        "0.756431312",
        "0.433926607"
      ],
      [
        "0.779824716",
        "0.623727160",
        "-0.878782257",
        "-0.255830614"
      ]
    ],
    [
      [
        "0.407178730",
        "0.330219151",
        "-0.884189622",
        "-0.106293881"
      ],
      [
        "0.873656041",
        "-0.527970734",
        "0.785813330",
        "-0.195353294"
      ]
    ],
    [
      [
        "0.907090387",
        "0.681590297",
        "-0.468562337",
        "0.432082767"
      ],
      [
        "0.731798253",
        "-0.780664533",
        "-0.866637204",
        "0.828314965"
      ]
    ],
    [
      [
        "0.199420267",
        "-0.484781897",
        "0.764885700",
        "0.975298565"
      ],
      [
        "0.538837199",
        "0.128198194",
        "-0.932356074",
        "0.235950002"
      ]
    ],
    [
      [
        "0.815431234",
        "0.728340932",
        "0.937117801",
        "-0.064333123"
      ],
      [
        "0.295666797",
        "0.590914743",
        "-0.234049318",
        "0.790939790"
      ]
    ],
    [
      [
        "0.407400743",
        "-0.260580431",
        "0.665125349",
        "0.644797139"
      ],
      [
        "0.693948242",
        "-0.238226633",
        "0.661739590",
        "-0.793224876"
      ]
    ],
    [
      [
        "0.341934839",
        "0.746595756",
        "-0.347521054",
        "-0.298860666"
      ],
      [
        "0.248922129",
        "0.764502617",
        "-0.271035704",
        "0.755327549"
      ]
    ]
  ]
}

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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