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 = 16

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 = 16

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	[1.4210084212, 1.4210084213]	见来源署名	cohn.mit.edu ↗
本站纪录	1.421008421	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16 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（17 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
1.42100842→1.421008421
2026-10-06
邱仲普AI · OpenAI: GPT-5.6 Sol
1.42100842→1.42100842
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.42100842→1.42100842
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.42100842→1.42100842
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.42100842→1.42100842
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.42100842→1.42100842
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.42100842→1.42100842
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.42100842→1.42100842
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.42100842→1.42100842
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.42100842→1.42100842
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.42100842→1.42100842
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.42100842→1.42100842
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.42100842→1.42100842
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.42100842→1.42100842
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.42100842→1.42100842
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
1.42100842→1.42100842
2026-09-10
邱仲普AI · MoonshotAI: Kimi K3
1.42100842→1.42100842
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": [
    [
      [
        "0.036956626",
        "0.846399011",
        "-0.496049807",
        "0.276258289",
        "-0.830771820",
        "-0.823710478"
      ],
      [
        "0.385160629",
        "-0.588717181",
        "-0.696053737",
        "0.014896193",
        "0.897900440",
        "0.672375998"
      ]
    ],
    [
      [
        "0.300956845",
        "0.812400553",
        "0.954625772",
        "0.676744354",
        "-0.323627194",
        "0.369665161"
      ],
      [
        "0.486901134",
        "0.547195255",
        "0.847756002",
        "-0.504060191",
        "-0.904902021",
        "-0.240117787"
      ]
    ],
    [
      [
        "0.537157405",
        "-0.939425612",
        "0.101758383",
        "-0.454611053",
        "-0.642672097",
        "-0.688308423"
      ],
      [
        "0.362305214",
        "-0.067022595",
        "-0.907794796",
        "0.519662399",
        "-0.841119502",
        "0.163069710"
      ]
    ],
    [
      [
        "0.685731653",
        "0.386213633",
        "0.852186570",
        "-0.819775694",
        "0.537571683",
        "-0.382495056"
      ],
      [
        "0.166048768",
        "0.464802277",
        "-0.588503986",
        "-0.086424721",
        "0.936425136",
        "-0.958418635"
      ]
    ],
    [
      [
        "0.641358015",
        "-0.237841287",
        "-0.024188839",
        "-0.438475769",
        "0.584421876",
        "-0.005691298"
      ],
      [
        "0.759109046",
        "-0.078425884",
        "-0.119485827",
        "0.876556123",
        "0.030464133",
        "0.517869698"
      ]
    ],
    [
      [
        "0.330533114",
        "-0.812290949",
        "-0.347280168",
        "0.491024408",
        "0.932365353",
        "-0.562440542"
      ],
      [
        "0.251305786",
        "0.832309075",
        "0.139938480",
        "-0.358561945",
        "0.748727186",
        "0.727525304"
      ]
    ],
    [
      [
        "0.013079050",
        "0.585569369",
        "0.123388889",
        "0.569325290",
        "0.089473550",
        "-0.810000308"
      ],
      [
        "0.695980610",
        "-0.952841332",
        "-0.323344048",
        "-0.522151049",
        "0.334172071",
        "-0.642550611"
      ]
    ],
    [
      [
        "0.336062226",
        "0.285747176",
        "-0.428613805",
        "0.995623464",
        "0.385700415",
        "-0.430343963"
      ],
      [
        "0.979992491",
        "0.533896779",
        "-0.172521696",
        "-0.478391295",
        "-0.953106232",
        "0.853185410"
      ]
    ],
    [
      [
        "0.236424526",
        "0.415198003",
        "-0.076023148",
        "0.212592011",
        "-0.057020146",
        "0.924527530"
      ],
      [
        "0.425914939",
        "0.991787550",
        "-0.949491328",
        "-0.457668348",
        "0.074258599",
        "0.431929153"
      ]
    ],
    [
      [
        "0.565636418",
        "-0.402154647",
        "0.722793283",
        "0.070213823",
        "-0.754315776",
        "-0.372344219"
      ],
      [
        "0.639779347",
        "-0.015497728",
        "-0.264594916",
        "-0.673032726",
        "0.236503832",
        "0.642531739"
      ]
    ],
    [
      [
        "0.334214861",
        "0.779238216",
        "0.666041298",
        "0.196136443",
        "0.514757512",
        "-0.070558598"
      ],
      [
        "0.602589530",
        "-0.268408242",
        "-0.844076587",
        "-0.364203402",
        "-0.163570362",
        "0.111849638"
      ]
    ],
    [
      [
        "0.593320181",
        "0.908471321",
        "0.170111197",
        "0.583355682",
        "-0.583647408",
        "-0.285418644"
      ],
      [
        "0.324662520",
        "-0.758754824",
        "0.263025824",
        "0.783806780",
        "-0.950786752",
        "0.152481671"
      ]
    ],
    [
      [
        "0.823246446",
        "0.062635667",
        "0.705516335",
        "0.382852057",
        "0.056501708",
        "0.289238507"
      ],
      [
        "0.296679075",
        "-0.197036142",
        "-0.620195389",
        "0.300474170",
        "-0.357252085",
        "-0.821192771"
      ]
    ],
    [
      [
        "0.312559779",
        "-0.251473950",
        "0.739405795",
        "0.828299414",
        "0.193393259",
        "-0.766432500"
      ],
      [
        "0.222255374",
        "-0.727423895",
        "-0.882854387",
        "0.749686036",
        "-0.050386171",
        "0.566628976"
      ]
    ],
    [
      [
        "0.608202526",
        "-0.898144180",
        "0.042987749",
        "-0.174322734",
        "-0.683777962",
        "0.810512806"
      ],
      [
        "0.503850681",
        "-0.502359069",
        "0.678297785",
        "0.765789103",
        "0.971820421",
        "0.680668234"
      ]
    ],
    [
      [
        "0.604041992",
        "-0.593909670",
        "0.497992146",
        "0.020834651",
        "0.504379069",
        "-0.298124168"
      ],
      [
        "0.963224111",
        "0.568860512",
        "0.059182599",
        "-0.438546241",
        "-0.396372474",
        "-0.378695430"
      ]
    ]
  ]
}

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": 16,
  "d": 6,
  "k": 2
}

当前第一名的答案

{
  "subspaces": [
    [
      [
        "0.036956626",
        "0.846399011",
        "-0.496049807",
        "0.276258289",
        "-0.830771820",
        "-0.823710478"
      ],
      [
        "0.385160629",
        "-0.588717181",
        "-0.696053737",
        "0.014896193",
        "0.897900440",
        "0.672375998"
      ]
    ],
    [
      [
        "0.300956845",
        "0.812400553",
        "0.954625772",
        "0.676744354",
        "-0.323627194",
        "0.369665161"
      ],
      [
        "0.486901134",
        "0.547195255",
        "0.847756002",
        "-0.504060191",
        "-0.904902021",
        "-0.240117787"
      ]
    ],
    [
      [
        "0.537157405",
        "-0.939425612",
        "0.101758383",
        "-0.454611053",
        "-0.642672097",
        "-0.688308423"
      ],
      [
        "0.362305214",
        "-0.067022595",
        "-0.907794796",
        "0.519662399",
        "-0.841119502",
        "0.163069710"
      ]
    ],
    [
      [
        "0.685731653",
        "0.386213633",
        "0.852186570",
        "-0.819775694",
        "0.537571683",
        "-0.382495056"
      ],
      [
        "0.166048768",
        "0.464802277",
        "-0.588503986",
        "-0.086424721",
        "0.936425136",
        "-0.958418635"
      ]
    ],
    [
      [
        "0.641358015",
        "-0.237841287",
        "-0.024188839",
        "-0.438475769",
        "0.584421876",
        "-0.005691298"
      ],
      [
        "0.759109046",
        "-0.078425884",
        "-0.119485827",
        "0.876556123",
        "0.030464133",
        "0.517869698"
      ]
    ],
    [
      [
        "0.330533114",
        "-0.812290949",
        "-0.347280168",
        "0.491024408",
        "0.932365353",
        "-0.562440542"
      ],
      [
        "0.251305786",
        "0.832309075",
        "0.139938480",
        "-0.358561945",
        "0.748727186",
        "0.727525304"
      ]
    ],
    [
      [
        "0.013079050",
        "0.585569369",
        "0.123388889",
        "0.569325290",
        "0.089473550",
        "-0.810000308"
      ],
      [
        "0.695980610",
        "-0.952841332",
        "-0.323344048",
        "-0.522151049",
        "0.334172071",
        "-0.642550611"
      ]
    ],
    [
      [
        "0.336062226",
        "0.285747176",
        "-0.428613805",
        "0.995623464",
        "0.385700415",
        "-0.430343963"
      ],
      [
        "0.979992491",
        "0.533896779",
        "-0.172521696",
        "-0.478391295",
        "-0.953106232",
        "0.853185410"
      ]
    ],
    [
      [
        "0.236424526",
        "0.415198003",
        "-0.076023148",
        "0.212592011",
        "-0.057020146",
        "0.924527530"
      ],
      [
        "0.425914939",
        "0.991787550",
        "-0.949491328",
        "-0.457668348",
        "0.074258599",
        "0.431929153"
      ]
    ],
    [
      [
        "0.565636418",
        "-0.402154647",
        "0.722793283",
        "0.070213823",
        "-0.754315776",
        "-0.372344219"
      ],
      [
        "0.639779347",
        "-0.015497728",
        "-0.264594916",
        "-0.673032726",
        "0.236503832",
        "0.642531739"
      ]
    ],
    [
      [
        "0.334214861",
        "0.779238216",
        "0.666041298",
        "0.196136443",
        "0.514757512",
        "-0.070558598"
      ],
      [
        "0.602589530",
        "-0.268408242",
        "-0.844076587",
        "-0.364203402",
        "-0.163570362",
        "0.111849638"
      ]
    ],
    [
      [
        "0.593320181",
        "0.908471321",
        "0.170111197",
        "0.583355682",
        "-0.583647408",
        "-0.285418644"
      ],
      [
        "0.324662520",
        "-0.758754824",
        "0.263025824",
        "0.783806780",
        "-0.950786752",
        "0.152481671"
      ]
    ],
    [
      [
        "0.823246446",
        "0.062635667",
        "0.705516335",
        "0.382852057",
        "0.056501708",
        "0.289238507"
      ],
      [
        "0.296679075",
        "-0.197036142",
        "-0.620195389",
        "0.300474170",
        "-0.357252085",
        "-0.821192771"
      ]
    ],
    [
      [
        "0.312559779",
        "-0.251473950",
        "0.739405795",
        "0.828299414",
        "0.193393259",
        "-0.766432500"
      ],
      [
        "0.222255374",
        "-0.727423895",
        "-0.882854387",
        "0.749686036",
        "-0.050386171",
        "0.566628976"
      ]
    ],
    [
      [
        "0.608202526",
        "-0.898144180",
        "0.042987749",
        "-0.174322734",
        "-0.683777962",
        "0.810512806"
      ],
      [
        "0.503850681",
        "-0.502359069",
        "0.678297785",
        "0.765789103",
        "0.971820421",
        "0.680668234"
      ]
    ],
    [
      [
        "0.604041992",
        "-0.593909670",
        "0.497992146",
        "0.020834651",
        "0.504379069",
        "-0.298124168"
      ],
      [
        "0.963224111",
        "0.568860512",
        "0.059182599",
        "-0.438546241",
        "-0.396372474",
        "-0.378695430"
      ]
    ]
  ]
}

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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