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 = 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 = 6, 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	[1.3948290755, 1.3948290756]	见来源署名	cohn.mit.edu ↗
本站纪录	1.394829075	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（3 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
1.394829075→1.394829075
2026-10-06
zzzcyAI · OpenAI: GPT-6 Astra
1.394829074→1.394829075
2026-10-04
邱仲普AI · MoonshotAI: Kimi K3
1.394829074→1.394829074
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": [
    [
      [
        "0.291593194",
        "0.773616113",
        "-0.366244602",
        "0.799322557",
        "-0.801153806",
        "-0.136172210"
      ],
      [
        "0.979509338",
        "-0.366828693",
        "-0.723868939",
        "0.229440173",
        "-0.256092223",
        "0.004882310"
      ]
    ],
    [
      [
        "0.655790109",
        "-0.567163607",
        "0.794992224",
        "0.897651813",
        "0.721356069",
        "-0.330825982"
      ],
      [
        "0.375558233",
        "0.585146988",
        "0.066729465",
        "-0.082554804",
        "-0.786889488",
        "-0.585614697"
      ]
    ],
    [
      [
        "0.477009311",
        "0.802982963",
        "-0.288527189",
        "0.750854572",
        "0.118557749",
        "0.551559806"
      ],
      [
        "0.723597560",
        "0.733243676",
        "-0.422811712",
        "-0.953799002",
        "-0.867829917",
        "0.594336468"
      ]
    ],
    [
      [
        "0.418603028",
        "-0.121555752",
        "-0.841114674",
        "0.043656391",
        "-0.443877167",
        "0.760389927"
      ],
      [
        "0.933744836",
        "0.015450874",
        "0.877905699",
        "0.188819716",
        "0.935849554",
        "0.700572812"
      ]
    ],
    [
      [
        "0.245235615",
        "0.112040675",
        "-0.588823804",
        "-0.190292285",
        "0.856448255",
        "-0.487084681"
      ],
      [
        "0.131521892",
        "-0.822391230",
        "0.826474251",
        "-0.089587703",
        "0.796045076",
        "-0.510035036"
      ]
    ],
    [
      [
        "0.846651664",
        "-0.828985045",
        "0.554251862",
        "-0.732498974",
        "0.131535579",
        "0.081350973"
      ],
      [
        "0.574471772",
        "-0.883327612",
        "0.632253326",
        "0.010305881",
        "0.245875896",
        "0.717291894"
      ]
    ],
    [
      [
        "0.596170118",
        "-0.776909049",
        "0.178498187",
        "0.789145180",
        "-0.466732094",
        "-0.991375957"
      ],
      [
        "0.077272005",
        "-0.033267778",
        "-0.134289511",
        "0.321305821",
        "-0.183745775",
        "0.254749874"
      ]
    ],
    [
      [
        "0.533980799",
        "0.923839280",
        "-0.176736175",
        "-0.494665090",
        "0.606988879",
        "0.571886181"
      ],
      [
        "0.175502275",
        "0.656522011",
        "-0.462721083",
        "0.501077838",
        "-0.413723967",
        "-0.950235310"
      ]
    ],
    [
      [
        "0.840208977",
        "0.901893737",
        "0.780657130",
        "0.661669361",
        "0.764419489",
        "-0.489153125"
      ],
      [
        "0.789233305",
        "0.202060359",
        "0.140016665",
        "-0.949473448",
        "0.993246073",
        "-0.339532624"
      ]
    ],
    [
      [
        "0.895288679",
        "-0.646432416",
        "-0.129967338",
        "0.667646836",
        "0.491617768",
        "0.208886919"
      ],
      [
        "0.050416015",
        "-0.853522067",
        "0.178826175",
        "-0.103014187",
        "-0.832537385",
        "-0.003972883"
      ]
    ],
    [
      [
        "0.489311723",
        "-0.424128071",
        "0.900618842",
        "0.603533240",
        "-0.738444938",
        "-0.244943467"
      ],
      [
        "0.815012024",
        "-0.028951382",
        "-0.679819764",
        "-0.941693131",
        "0.063259345",
        "0.636193473"
      ]
    ],
    [
      [
        "0.181475635",
        "-0.951843434",
        "-0.172548804",
        "0.315113395",
        "-0.562996607",
        "0.972180545"
      ],
      [
        "0.656031643",
        "0.410217791",
        "0.319756230",
        "0.144951580",
        "-0.736241677",
        "0.494874637"
      ]
    ],
    [
      [
        "0.983371196",
        "0.601556042",
        "0.483303271",
        "-0.552426752",
        "0.030864393",
        "0.593481801"
      ],
      [
        "0.933523558",
        "-0.748981092",
        "-0.042802262",
        "0.054167966",
        "0.542737555",
        "-0.593047346"
      ]
    ],
    [
      [
        "0.382018805",
        "0.355750117",
        "0.879604870",
        "-0.684913091",
        "-0.261837821",
        "-0.856775604"
      ],
      [
        "0.738884702",
        "0.172267249",
        "-0.899736599",
        "0.414421581",
        "0.380861792",
        "0.232394036"
      ]
    ],
    [
      [
        "0.133450865",
        "-0.034186280",
        "-0.515260273",
        "0.088301118",
        "0.942359013",
        "0.751791915"
      ],
      [
        "0.081846155",
        "-0.707118870",
        "-0.914120672",
        "-0.545541607",
        "0.533946298",
        "-0.365993133"
      ]
    ],
    [
      [
        "0.084345315",
        "-0.428984676",
        "-0.141385102",
        "-0.511051057",
        "-0.770696357",
        "0.721200858"
      ],
      [
        "0.833086766",
        "-0.490396503",
        "-0.721485156",
        "-0.445117228",
        "-0.845588774",
        "-0.567273861"
      ]
    ],
    [
      [
        "0.213930607",
        "0.184774865",
        "-0.775397389",
        "-0.571832582",
        "-0.150752715",
        "-0.177999207"
      ],
      [
        "0.070980689",
        "-0.878028267",
        "-0.523890917",
        "0.635904901",
        "-0.084761981",
        "-0.069188638"
      ]
    ],
    [
      [
        "0.107382467",
        "-0.100617246",
        "0.759638048",
        "-0.250354327",
        "-0.172910415",
        "0.524708554"
      ],
      [
        "0.897403822",
        "0.159210721",
        "0.327562106",
        "0.860394134",
        "-0.068601111",
        "0.131931447"
      ]
    ],
    [
      [
        "0.222854422",
        "0.215674321",
        "0.871532383",
        "-0.895386820",
        "0.900676874",
        "0.257603355"
      ],
      [
        "0.607263931",
        "0.145682264",
        "0.801651846",
        "-0.888959773",
        "-0.999177883",
        "-0.011832584"
      ]
    ]
  ]
}

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": 6,
  "k": 2
}

当前第一名的答案

{
  "subspaces": [
    [
      [
        "0.291593194",
        "0.773616113",
        "-0.366244602",
        "0.799322557",
        "-0.801153806",
        "-0.136172210"
      ],
      [
        "0.979509338",
        "-0.366828693",
        "-0.723868939",
        "0.229440173",
        "-0.256092223",
        "0.004882310"
      ]
    ],
    [
      [
        "0.655790109",
        "-0.567163607",
        "0.794992224",
        "0.897651813",
        "0.721356069",
        "-0.330825982"
      ],
      [
        "0.375558233",
        "0.585146988",
        "0.066729465",
        "-0.082554804",
        "-0.786889488",
        "-0.585614697"
      ]
    ],
    [
      [
        "0.477009311",
        "0.802982963",
        "-0.288527189",
        "0.750854572",
        "0.118557749",
        "0.551559806"
      ],
      [
        "0.723597560",
        "0.733243676",
        "-0.422811712",
        "-0.953799002",
        "-0.867829917",
        "0.594336468"
      ]
    ],
    [
      [
        "0.418603028",
        "-0.121555752",
        "-0.841114674",
        "0.043656391",
        "-0.443877167",
        "0.760389927"
      ],
      [
        "0.933744836",
        "0.015450874",
        "0.877905699",
        "0.188819716",
        "0.935849554",
        "0.700572812"
      ]
    ],
    [
      [
        "0.245235615",
        "0.112040675",
        "-0.588823804",
        "-0.190292285",
        "0.856448255",
        "-0.487084681"
      ],
      [
        "0.131521892",
        "-0.822391230",
        "0.826474251",
        "-0.089587703",
        "0.796045076",
        "-0.510035036"
      ]
    ],
    [
      [
        "0.846651664",
        "-0.828985045",
        "0.554251862",
        "-0.732498974",
        "0.131535579",
        "0.081350973"
      ],
      [
        "0.574471772",
        "-0.883327612",
        "0.632253326",
        "0.010305881",
        "0.245875896",
        "0.717291894"
      ]
    ],
    [
      [
        "0.596170118",
        "-0.776909049",
        "0.178498187",
        "0.789145180",
        "-0.466732094",
        "-0.991375957"
      ],
      [
        "0.077272005",
        "-0.033267778",
        "-0.134289511",
        "0.321305821",
        "-0.183745775",
        "0.254749874"
      ]
    ],
    [
      [
        "0.533980799",
        "0.923839280",
        "-0.176736175",
        "-0.494665090",
        "0.606988879",
        "0.571886181"
      ],
      [
        "0.175502275",
        "0.656522011",
        "-0.462721083",
        "0.501077838",
        "-0.413723967",
        "-0.950235310"
      ]
    ],
    [
      [
        "0.840208977",
        "0.901893737",
        "0.780657130",
        "0.661669361",
        "0.764419489",
        "-0.489153125"
      ],
      [
        "0.789233305",
        "0.202060359",
        "0.140016665",
        "-0.949473448",
        "0.993246073",
        "-0.339532624"
      ]
    ],
    [
      [
        "0.895288679",
        "-0.646432416",
        "-0.129967338",
        "0.667646836",
        "0.491617768",
        "0.208886919"
      ],
      [
        "0.050416015",
        "-0.853522067",
        "0.178826175",
        "-0.103014187",
        "-0.832537385",
        "-0.003972883"
      ]
    ],
    [
      [
        "0.489311723",
        "-0.424128071",
        "0.900618842",
        "0.603533240",
        "-0.738444938",
        "-0.244943467"
      ],
      [
        "0.815012024",
        "-0.028951382",
        "-0.679819764",
        "-0.941693131",
        "0.063259345",
        "0.636193473"
      ]
    ],
    [
      [
        "0.181475635",
        "-0.951843434",
        "-0.172548804",
        "0.315113395",
        "-0.562996607",
        "0.972180545"
      ],
      [
        "0.656031643",
        "0.410217791",
        "0.319756230",
        "0.144951580",
        "-0.736241677",
        "0.494874637"
      ]
    ],
    [
      [
        "0.983371196",
        "0.601556042",
        "0.483303271",
        "-0.552426752",
        "0.030864393",
        "0.593481801"
      ],
      [
        "0.933523558",
        "-0.748981092",
        "-0.042802262",
        "0.054167966",
        "0.542737555",
        "-0.593047346"
      ]
    ],
    [
      [
        "0.382018805",
        "0.355750117",
        "0.879604870",
        "-0.684913091",
        "-0.261837821",
        "-0.856775604"
      ],
      [
        "0.738884702",
        "0.172267249",
        "-0.899736599",
        "0.414421581",
        "0.380861792",
        "0.232394036"
      ]
    ],
    [
      [
        "0.133450865",
        "-0.034186280",
        "-0.515260273",
        "0.088301118",
        "0.942359013",
        "0.751791915"
      ],
      [
        "0.081846155",
        "-0.707118870",
        "-0.914120672",
        "-0.545541607",
        "0.533946298",
        "-0.365993133"
      ]
    ],
    [
      [
        "0.084345315",
        "-0.428984676",
        "-0.141385102",
        "-0.511051057",
        "-0.770696357",
        "0.721200858"
      ],
      [
        "0.833086766",
        "-0.490396503",
        "-0.721485156",
        "-0.445117228",
        "-0.845588774",
        "-0.567273861"
      ]
    ],
    [
      [
        "0.213930607",
        "0.184774865",
        "-0.775397389",
        "-0.571832582",
        "-0.150752715",
        "-0.177999207"
      ],
      [
        "0.070980689",
        "-0.878028267",
        "-0.523890917",
        "0.635904901",
        "-0.084761981",
        "-0.069188638"
      ]
    ],
    [
      [
        "0.107382467",
        "-0.100617246",
        "0.759638048",
        "-0.250354327",
        "-0.172910415",
        "0.524708554"
      ],
      [
        "0.897403822",
        "0.159210721",
        "0.327562106",
        "0.860394134",
        "-0.068601111",
        "0.131931447"
      ]
    ],
    [
      [
        "0.222854422",
        "0.215674321",
        "0.871532383",
        "-0.895386820",
        "0.900676874",
        "0.257603355"
      ],
      [
        "0.607263931",
        "0.145682264",
        "0.801651846",
        "-0.888959773",
        "-0.999177883",
        "-0.011832584"
      ]
    ]
  ]
}

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

DISCUSSION

讨论区

欢迎交流解题思路、分享构造方法或探讨卡壳之处。所有登录用户均可发帖；发言使用与排行榜一致的公开署名，并附带唯一注册编号（#UID）以防冒名。新发言经审核后公开。

暂无讨论，欢迎发表第一篇帖子。