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 = 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 = 4, 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	[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（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": [
    [
      [
        "0.605472138",
        "0.003098643",
        "0.319397680",
        "0.534452822"
      ],
      [
        "0.656428275",
        "-0.641320247",
        "-0.901440476",
        "-0.957086468"
      ]
    ],
    [
      [
        "0.513131811",
        "-0.294137416",
        "-0.542857019",
        "-0.528220046"
      ],
      [
        "0.949172948",
        "0.297797580",
        "0.558490487",
        "-0.888339341"
      ]
    ],
    [
      [
        "0.436311327",
        "-0.445761327",
        "0.405305964",
        "-0.877263362"
      ],
      [
        "0.133380234",
        "-0.611372439",
        "-0.297171742",
        "0.431473835"
      ]
    ],
    [
      [
        "0.196442025",
        "0.132319590",
        "0.453412381",
        "-0.143334870"
      ],
      [
        "0.515094527",
        "-0.673061363",
        "-0.854407389",
        "0.713663112"
      ]
    ],
    [
      [
        "0.474832295",
        "0.035824737",
        "0.178726350",
        "-0.752863030"
      ],
      [
        "0.948883446",
        "0.054559844",
        "-0.455388767",
        "0.611263204"
      ]
    ],
    [
      [
        "0.397969084",
        "-0.959152773",
        "0.579720890",
        "-0.324978429"
      ],
      [
        "0.580777627",
        "0.246352338",
        "-0.173639994",
        "0.657598187"
      ]
    ],
    [
      [
        "0.649647009",
        "0.845496088",
        "-0.213245902",
        "-0.580909905"
      ],
      [
        "0.828613288",
        "-0.562906470",
        "0.464985088",
        "0.756762807"
      ]
    ],
    [
      [
        "0.403935035",
        "0.797330969",
        "0.557531435",
        "-0.684297647"
      ],
      [
        "0.139458364",
        "0.873078345",
        "-0.737825646",
        "-0.336734273"
      ]
    ],
    [
      [
        "0.263887234",
        "-0.097104833",
        "0.413752109",
        "0.957275127"
      ],
      [
        "0.037456731",
        "-0.954809934",
        "-0.321411796",
        "0.864554152"
      ]
    ],
    [
      [
        "0.900151240",
        "-0.937072786",
        "-0.031385252",
        "-0.694269719"
      ],
      [
        "0.624728372",
        "0.703358614",
        "0.042327691",
        "0.067781460"
      ]
    ],
    [
      [
        "0.265832494",
        "0.279350078",
        "0.875926337",
        "0.255529857"
      ],
      [
        "0.300401912",
        "-0.537533543",
        "0.275346586",
        "-0.110023270"
      ]
    ],
    [
      [
        "0.480629063",
        "0.309955945",
        "-0.857849000",
        "0.483822931"
      ],
      [
        "0.040608634",
        "0.829689126",
        "-0.335611825",
        "0.919532059"
      ]
    ],
    [
      [
        "0.615703401",
        "0.444737405",
        "-0.940456615",
        "-0.323122414"
      ],
      [
        "0.580857825",
        "0.757590863",
        "0.859687183",
        "0.553200819"
      ]
    ],
    [
      [
        "0.407178730",
        "0.330219151",
        "-0.884189622",
        "-0.106293881"
      ],
      [
        "0.873656041",
        "-0.527970734",
        "0.785813330",
        "-0.195353294"
      ]
    ],
    [
      [
        "0.143233597",
        "0.744194091",
        "0.159782993",
        "-0.161288988"
      ],
      [
        "0.963051086",
        "-0.068375729",
        "-0.788326063",
        "0.744302806"
      ]
    ],
    [
      [
        "0.138068454",
        "0.313797322",
        "-0.824811693",
        "-0.408279875"
      ],
      [
        "0.814848966",
        "-0.091473021",
        "-0.814876082",
        "0.832875307"
      ]
    ],
    [
      [
        "0.276824241",
        "-0.131076833",
        "0.982413003",
        "-0.964438934"
      ],
      [
        "0.846907034",
        "0.962618415",
        "0.611310732",
        "0.446826679"
      ]
    ],
    [
      [
        "0.503648786",
        "-0.128169708",
        "0.377778101",
        "-0.987141416"
      ],
      [
        "0.408425912",
        "-0.280422446",
        "0.710763885",
        "0.822906271"
      ]
    ],
    [
      [
        "0.378896223",
        "0.841335349",
        "-0.386232526",
        "-0.269372836"
      ],
      [
        "0.110349638",
        "0.460680837",
        "-0.130096741",
        "0.870912578"
      ]
    ]
  ]
}

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": 4,
  "k": 2
}

当前第一名的答案

{
  "subspaces": [
    [
      [
        "0.605472138",
        "0.003098643",
        "0.319397680",
        "0.534452822"
      ],
      [
        "0.656428275",
        "-0.641320247",
        "-0.901440476",
        "-0.957086468"
      ]
    ],
    [
      [
        "0.513131811",
        "-0.294137416",
        "-0.542857019",
        "-0.528220046"
      ],
      [
        "0.949172948",
        "0.297797580",
        "0.558490487",
        "-0.888339341"
      ]
    ],
    [
      [
        "0.436311327",
        "-0.445761327",
        "0.405305964",
        "-0.877263362"
      ],
      [
        "0.133380234",
        "-0.611372439",
        "-0.297171742",
        "0.431473835"
      ]
    ],
    [
      [
        "0.196442025",
        "0.132319590",
        "0.453412381",
        "-0.143334870"
      ],
      [
        "0.515094527",
        "-0.673061363",
        "-0.854407389",
        "0.713663112"
      ]
    ],
    [
      [
        "0.474832295",
        "0.035824737",
        "0.178726350",
        "-0.752863030"
      ],
      [
        "0.948883446",
        "0.054559844",
        "-0.455388767",
        "0.611263204"
      ]
    ],
    [
      [
        "0.397969084",
        "-0.959152773",
        "0.579720890",
        "-0.324978429"
      ],
      [
        "0.580777627",
        "0.246352338",
        "-0.173639994",
        "0.657598187"
      ]
    ],
    [
      [
        "0.649647009",
        "0.845496088",
        "-0.213245902",
        "-0.580909905"
      ],
      [
        "0.828613288",
        "-0.562906470",
        "0.464985088",
        "0.756762807"
      ]
    ],
    [
      [
        "0.403935035",
        "0.797330969",
        "0.557531435",
        "-0.684297647"
      ],
      [
        "0.139458364",
        "0.873078345",
        "-0.737825646",
        "-0.336734273"
      ]
    ],
    [
      [
        "0.263887234",
        "-0.097104833",
        "0.413752109",
        "0.957275127"
      ],
      [
        "0.037456731",
        "-0.954809934",
        "-0.321411796",
        "0.864554152"
      ]
    ],
    [
      [
        "0.900151240",
        "-0.937072786",
        "-0.031385252",
        "-0.694269719"
      ],
      [
        "0.624728372",
        "0.703358614",
        "0.042327691",
        "0.067781460"
      ]
    ],
    [
      [
        "0.265832494",
        "0.279350078",
        "0.875926337",
        "0.255529857"
      ],
      [
        "0.300401912",
        "-0.537533543",
        "0.275346586",
        "-0.110023270"
      ]
    ],
    [
      [
        "0.480629063",
        "0.309955945",
        "-0.857849000",
        "0.483822931"
      ],
      [
        "0.040608634",
        "0.829689126",
        "-0.335611825",
        "0.919532059"
      ]
    ],
    [
      [
        "0.615703401",
        "0.444737405",
        "-0.940456615",
        "-0.323122414"
      ],
      [
        "0.580857825",
        "0.757590863",
        "0.859687183",
        "0.553200819"
      ]
    ],
    [
      [
        "0.407178730",
        "0.330219151",
        "-0.884189622",
        "-0.106293881"
      ],
      [
        "0.873656041",
        "-0.527970734",
        "0.785813330",
        "-0.195353294"
      ]
    ],
    [
      [
        "0.143233597",
        "0.744194091",
        "0.159782993",
        "-0.161288988"
      ],
      [
        "0.963051086",
        "-0.068375729",
        "-0.788326063",
        "0.744302806"
      ]
    ],
    [
      [
        "0.138068454",
        "0.313797322",
        "-0.824811693",
        "-0.408279875"
      ],
      [
        "0.814848966",
        "-0.091473021",
        "-0.814876082",
        "0.832875307"
      ]
    ],
    [
      [
        "0.276824241",
        "-0.131076833",
        "0.982413003",
        "-0.964438934"
      ],
      [
        "0.846907034",
        "0.962618415",
        "0.611310732",
        "0.446826679"
      ]
    ],
    [
      [
        "0.503648786",
        "-0.128169708",
        "0.377778101",
        "-0.987141416"
      ],
      [
        "0.408425912",
        "-0.280422446",
        "0.710763885",
        "0.822906271"
      ]
    ],
    [
      [
        "0.378896223",
        "0.841335349",
        "-0.386232526",
        "-0.269372836"
      ],
      [
        "0.110349638",
        "0.460680837",
        "-0.130096741",
        "0.870912578"
      ]
    ]
  ]
}

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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