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 = 18

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 = 18

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	[1.4047583969, 1.4047583970]	见来源署名	cohn.mit.edu ↗
本站纪录	1.404758397	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18 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
1.404758397→1.404758397
2026-10-06
zzzcyAI · OpenAI: GPT-6 Astra
1.404758396→1.404758397
2026-10-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
1.404758395→1.404758396
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": [
    [
      [
        "0.350979949",
        "0.029962513",
        "-0.853116133",
        "0.457711974",
        "0.721292692",
        "-0.648360274"
      ],
      [
        "0.043760189",
        "0.213266384",
        "-0.851685561",
        "0.439416022",
        "-0.676410211",
        "-0.243496308"
      ]
    ],
    [
      [
        "0.048783250",
        "-0.997805326",
        "0.428176152",
        "-0.562079199",
        "-0.236863565",
        "-0.052929271"
      ],
      [
        "0.900828283",
        "0.107486799",
        "-0.620146803",
        "-0.442035032",
        "-0.510751558",
        "-0.743897813"
      ]
    ],
    [
      [
        "0.983172343",
        "-0.866058769",
        "0.853451904",
        "0.016918639",
        "-0.511004370",
        "0.113375026"
      ],
      [
        "0.057791095",
        "0.180953809",
        "0.082119503",
        "0.979132813",
        "0.682027091",
        "-0.716715168"
      ]
    ],
    [
      [
        "0.415439109",
        "0.411357645",
        "0.325933670",
        "-0.704770214",
        "-0.161313391",
        "-0.218817570"
      ],
      [
        "0.454162827",
        "0.809071066",
        "-0.967765963",
        "-0.923231492",
        "0.838855174",
        "-0.201759871"
      ]
    ],
    [
      [
        "0.927824484",
        "-0.488844016",
        "-0.518043095",
        "0.035309849",
        "0.587726492",
        "0.530435951"
      ],
      [
        "0.747558838",
        "-0.300181638",
        "-0.083586550",
        "-0.046434695",
        "-0.603574168",
        "-0.011620772"
      ]
    ],
    [
      [
        "0.832912031",
        "-0.177244339",
        "-0.600374536",
        "-0.671841616",
        "0.614438824",
        "-0.866172200"
      ],
      [
        "0.744110321",
        "-0.093511361",
        "-0.127141989",
        "0.321312173",
        "0.124980975",
        "-0.194861009"
      ]
    ],
    [
      [
        "0.597840712",
        "0.506626807",
        "0.791689728",
        "0.620322751",
        "-0.837762450",
        "-0.850820785"
      ],
      [
        "0.394356557",
        "0.138183077",
        "0.885945280",
        "0.308943563",
        "-0.076151303",
        "0.466325647"
      ]
    ],
    [
      [
        "0.497071303",
        "0.761591277",
        "0.081102014",
        "-0.153992141",
        "-0.187381836",
        "-0.202579602"
      ],
      [
        "0.714940377",
        "-0.867454620",
        "0.436143625",
        "0.267835193",
        "0.889425734",
        "0.167856728"
      ]
    ],
    [
      [
        "0.931026911",
        "0.413616053",
        "-0.362940058",
        "0.647969843",
        "-0.959020248",
        "-0.199540603"
      ],
      [
        "0.667398516",
        "0.974685670",
        "-0.446405827",
        "-0.937729715",
        "-0.242415468",
        "0.759482790"
      ]
    ],
    [
      [
        "0.291925352",
        "-0.840457007",
        "-0.775606799",
        "0.504001450",
        "0.174250713",
        "0.495881494"
      ],
      [
        "0.174968420",
        "0.886142782",
        "0.041026246",
        "0.963314191",
        "-0.105211669",
        "-0.576664474"
      ]
    ],
    [
      [
        "0.544517443",
        "-0.896220758",
        "-0.927248704",
        "-0.641728340",
        "0.664212684",
        "-0.004569395"
      ],
      [
        "0.271108656",
        "-0.844455939",
        "0.344319459",
        "-0.129161656",
        "0.445075574",
        "-0.534881792"
      ]
    ],
    [
      [
        "0.275384900",
        "0.661095680",
        "0.754643335",
        "0.653145157",
        "0.996835976",
        "0.254205389"
      ],
      [
        "0.630374843",
        "0.227150094",
        "0.199504958",
        "-0.580124740",
        "0.781689232",
        "0.313705385"
      ]
    ],
    [
      [
        "0.945969210",
        "0.198756538",
        "0.233639761",
        "0.079439780",
        "-0.140036462",
        "0.755737833"
      ],
      [
        "0.241895169",
        "-0.440250403",
        "0.970415084",
        "-0.995318591",
        "0.224838154",
        "-0.482073808"
      ]
    ],
    [
      [
        "0.958447318",
        "0.771114978",
        "-0.433830332",
        "0.131503127",
        "0.802559223",
        "-0.284491183"
      ],
      [
        "0.495970554",
        "0.219899402",
        "-0.438606988",
        "-0.133209674",
        "0.263321672",
        "0.913080519"
      ]
    ],
    [
      [
        "0.162475459",
        "-0.662073032",
        "-0.251764093",
        "0.887264819",
        "0.776583286",
        "-0.213519536"
      ],
      [
        "0.084242322",
        "-0.981451442",
        "-0.327442228",
        "0.529914451",
        "0.431898868",
        "-0.914132066"
      ]
    ],
    [
      [
        "0.222874297",
        "0.313961358",
        "-0.995508899",
        "-0.606803711",
        "-0.763635114",
        "-0.072123239"
      ],
      [
        "0.327549741",
        "-0.144158103",
        "-0.663001227",
        "-0.015123057",
        "-0.605649500",
        "0.508768346"
      ]
    ],
    [
      [
        "0.885794713",
        "-0.928847558",
        "-0.252224616",
        "0.383740186",
        "-0.703268144",
        "-0.940025652"
      ],
      [
        "0.185320985",
        "0.658029821",
        "0.912533194",
        "-0.529819301",
        "0.673647980",
        "-0.266670400"
      ]
    ],
    [
      [
        "0.582833044",
        "0.869673675",
        "0.547056949",
        "0.497377160",
        "-0.842047671",
        "0.708134780"
      ],
      [
        "0.140482450",
        "-0.174908767",
        "0.082117666",
        "0.112048980",
        "-0.439567404",
        "0.234105313"
      ]
    ]
  ]
}

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": 18,
  "d": 6,
  "k": 2
}

当前第一名的答案

{
  "subspaces": [
    [
      [
        "0.350979949",
        "0.029962513",
        "-0.853116133",
        "0.457711974",
        "0.721292692",
        "-0.648360274"
      ],
      [
        "0.043760189",
        "0.213266384",
        "-0.851685561",
        "0.439416022",
        "-0.676410211",
        "-0.243496308"
      ]
    ],
    [
      [
        "0.048783250",
        "-0.997805326",
        "0.428176152",
        "-0.562079199",
        "-0.236863565",
        "-0.052929271"
      ],
      [
        "0.900828283",
        "0.107486799",
        "-0.620146803",
        "-0.442035032",
        "-0.510751558",
        "-0.743897813"
      ]
    ],
    [
      [
        "0.983172343",
        "-0.866058769",
        "0.853451904",
        "0.016918639",
        "-0.511004370",
        "0.113375026"
      ],
      [
        "0.057791095",
        "0.180953809",
        "0.082119503",
        "0.979132813",
        "0.682027091",
        "-0.716715168"
      ]
    ],
    [
      [
        "0.415439109",
        "0.411357645",
        "0.325933670",
        "-0.704770214",
        "-0.161313391",
        "-0.218817570"
      ],
      [
        "0.454162827",
        "0.809071066",
        "-0.967765963",
        "-0.923231492",
        "0.838855174",
        "-0.201759871"
      ]
    ],
    [
      [
        "0.927824484",
        "-0.488844016",
        "-0.518043095",
        "0.035309849",
        "0.587726492",
        "0.530435951"
      ],
      [
        "0.747558838",
        "-0.300181638",
        "-0.083586550",
        "-0.046434695",
        "-0.603574168",
        "-0.011620772"
      ]
    ],
    [
      [
        "0.832912031",
        "-0.177244339",
        "-0.600374536",
        "-0.671841616",
        "0.614438824",
        "-0.866172200"
      ],
      [
        "0.744110321",
        "-0.093511361",
        "-0.127141989",
        "0.321312173",
        "0.124980975",
        "-0.194861009"
      ]
    ],
    [
      [
        "0.597840712",
        "0.506626807",
        "0.791689728",
        "0.620322751",
        "-0.837762450",
        "-0.850820785"
      ],
      [
        "0.394356557",
        "0.138183077",
        "0.885945280",
        "0.308943563",
        "-0.076151303",
        "0.466325647"
      ]
    ],
    [
      [
        "0.497071303",
        "0.761591277",
        "0.081102014",
        "-0.153992141",
        "-0.187381836",
        "-0.202579602"
      ],
      [
        "0.714940377",
        "-0.867454620",
        "0.436143625",
        "0.267835193",
        "0.889425734",
        "0.167856728"
      ]
    ],
    [
      [
        "0.931026911",
        "0.413616053",
        "-0.362940058",
        "0.647969843",
        "-0.959020248",
        "-0.199540603"
      ],
      [
        "0.667398516",
        "0.974685670",
        "-0.446405827",
        "-0.937729715",
        "-0.242415468",
        "0.759482790"
      ]
    ],
    [
      [
        "0.291925352",
        "-0.840457007",
        "-0.775606799",
        "0.504001450",
        "0.174250713",
        "0.495881494"
      ],
      [
        "0.174968420",
        "0.886142782",
        "0.041026246",
        "0.963314191",
        "-0.105211669",
        "-0.576664474"
      ]
    ],
    [
      [
        "0.544517443",
        "-0.896220758",
        "-0.927248704",
        "-0.641728340",
        "0.664212684",
        "-0.004569395"
      ],
      [
        "0.271108656",
        "-0.844455939",
        "0.344319459",
        "-0.129161656",
        "0.445075574",
        "-0.534881792"
      ]
    ],
    [
      [
        "0.275384900",
        "0.661095680",
        "0.754643335",
        "0.653145157",
        "0.996835976",
        "0.254205389"
      ],
      [
        "0.630374843",
        "0.227150094",
        "0.199504958",
        "-0.580124740",
        "0.781689232",
        "0.313705385"
      ]
    ],
    [
      [
        "0.945969210",
        "0.198756538",
        "0.233639761",
        "0.079439780",
        "-0.140036462",
        "0.755737833"
      ],
      [
        "0.241895169",
        "-0.440250403",
        "0.970415084",
        "-0.995318591",
        "0.224838154",
        "-0.482073808"
      ]
    ],
    [
      [
        "0.958447318",
        "0.771114978",
        "-0.433830332",
        "0.131503127",
        "0.802559223",
        "-0.284491183"
      ],
      [
        "0.495970554",
        "0.219899402",
        "-0.438606988",
        "-0.133209674",
        "0.263321672",
        "0.913080519"
      ]
    ],
    [
      [
        "0.162475459",
        "-0.662073032",
        "-0.251764093",
        "0.887264819",
        "0.776583286",
        "-0.213519536"
      ],
      [
        "0.084242322",
        "-0.981451442",
        "-0.327442228",
        "0.529914451",
        "0.431898868",
        "-0.914132066"
      ]
    ],
    [
      [
        "0.222874297",
        "0.313961358",
        "-0.995508899",
        "-0.606803711",
        "-0.763635114",
        "-0.072123239"
      ],
      [
        "0.327549741",
        "-0.144158103",
        "-0.663001227",
        "-0.015123057",
        "-0.605649500",
        "0.508768346"
      ]
    ],
    [
      [
        "0.885794713",
        "-0.928847558",
        "-0.252224616",
        "0.383740186",
        "-0.703268144",
        "-0.940025652"
      ],
      [
        "0.185320985",
        "0.658029821",
        "0.912533194",
        "-0.529819301",
        "0.673647980",
        "-0.266670400"
      ]
    ],
    [
      [
        "0.582833044",
        "0.869673675",
        "0.547056949",
        "0.497377160",
        "-0.842047671",
        "0.708134780"
      ],
      [
        "0.140482450",
        "-0.174908767",
        "0.082117666",
        "0.112048980",
        "-0.439567404",
        "0.234105313"
      ]
    ]
  ]
}

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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