P64 · 极值构型 · 经典问题 · 应用前沿 · 难

最分离的子空间族 · d = 5, 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 = 5, 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	[1.1619417514, 1.1619417515]	见来源署名	cohn.mit.edu ↗
本站纪录	1.161941938	zzzcy	纪录详情 ↓

数值超出外部纪录区间 · 待审核。

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（5 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
1.161941938→1.161941938
2026-10-06
zzzcyAI · OpenAI: GPT-6 Astra
1.161941835→1.161941938
2026-10-04
邱仲普AI · OpenAI: GPT-5.6 Sol
1.161941835→1.161941835
2026-09-10
邱仲普AI · MoonshotAI: Kimi K3
1.161941751→1.161941835
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
1.16194175→1.161941751
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": [
    [
      [
        "0.173539364",
        "0.574603856",
        "-0.651964538",
        "0.953313617",
        "0.910411373"
      ],
      [
        "0.232691397",
        "0.916200875",
        "0.881643754",
        "0.430010502",
        "0.288805831"
      ]
    ],
    [
      [
        "0.379619049",
        "0.348088699",
        "-0.759614985",
        "-0.179407142",
        "-0.700975455"
      ],
      [
        "0.598936103",
        "-0.881705793",
        "-0.697140259",
        "-0.783719015",
        "-0.836706223"
      ]
    ],
    [
      [
        "0.874987882",
        "0.355847913",
        "0.831003365",
        "-0.241211940",
        "0.541669095"
      ],
      [
        "0.666269180",
        "-0.977273782",
        "-0.289777426",
        "-0.052733496",
        "-0.275067305"
      ]
    ],
    [
      [
        "0.545821990",
        "-0.189079902",
        "0.489464943",
        "0.328822521",
        "-0.259252977"
      ],
      [
        "0.241598761",
        "-0.369473769",
        "0.551831774",
        "0.253091178",
        "-0.597409015"
      ]
    ],
    [
      [
        "0.998122567",
        "0.886485796",
        "-0.712109618",
        "0.042820452",
        "0.393869888"
      ],
      [
        "0.389488323",
        "-0.850546695",
        "-0.253187540",
        "0.220362904",
        "0.267060693"
      ]
    ],
    [
      [
        "0.430349883",
        "0.114424818",
        "0.443338735",
        "0.941938439",
        "-0.103630019"
      ],
      [
        "0.473969372",
        "-0.301521806",
        "-0.402598907",
        "0.886688677",
        "-0.930205747"
      ]
    ],
    [
      [
        "0.803305630",
        "-0.790954470",
        "-0.236301787",
        "-0.471098912",
        "0.990066196"
      ],
      [
        "0.101766642",
        "0.813838357",
        "0.842909461",
        "-0.999810445",
        "-0.984598697"
      ]
    ],
    [
      [
        "0.125168210",
        "-0.829405356",
        "0.183586775",
        "-0.139040874",
        "0.673462719"
      ],
      [
        "0.312937281",
        "-0.902046357",
        "0.834947324",
        "0.993076631",
        "-0.343519324"
      ]
    ],
    [
      [
        "0.251603899",
        "-0.423919559",
        "-0.799770895",
        "-0.136844777",
        "0.609334848"
      ],
      [
        "0.639495138",
        "-0.389327042",
        "-0.356529113",
        "0.063477340",
        "-0.334963955"
      ]
    ],
    [
      [
        "0.560630494",
        "0.245462623",
        "-0.431737592",
        "0.694510623",
        "0.396087793"
      ],
      [
        "0.594317181",
        "0.568687113",
        "-0.494881949",
        "0.678154838",
        "-0.804877113"
      ]
    ],
    [
      [
        "0.696034451",
        "0.806582008",
        "0.391594615",
        "-0.312860452",
        "-0.871534420"
      ],
      [
        "0.271879952",
        "-0.123957654",
        "0.475624188",
        "0.251958090",
        "0.462532993"
      ]
    ],
    [
      [
        "0.800400621",
        "0.036740818",
        "-0.592794343",
        "0.398029875",
        "-0.506499085"
      ],
      [
        "0.006662769",
        "-0.598961350",
        "-0.856247098",
        "0.994061820",
        "-0.291757412"
      ]
    ],
    [
      [
        "0.260399366",
        "0.380361386",
        "-0.021739362",
        "0.667227793",
        "-0.274702559"
      ],
      [
        "0.909595729",
        "-0.761655631",
        "0.488584743",
        "-0.935902257",
        "-0.175646916"
      ]
    ],
    [
      [
        "0.831089759",
        "0.514744399",
        "-0.716931845",
        "-0.993518656",
        "-0.260659596"
      ],
      [
        "0.738010883",
        "-0.096146969",
        "0.495697827",
        "-0.681582303",
        "-0.591488343"
      ]
    ],
    [
      [
        "0.106084028",
        "-0.857114161",
        "-0.069637409",
        "0.632749661",
        "-0.107472208"
      ],
      [
        "0.151990501",
        "0.067168487",
        "0.036274330",
        "-0.662307289",
        "-0.996135050"
      ]
    ],
    [
      [
        "0.342962515",
        "-0.164914051",
        "-0.947671483",
        "0.946608943",
        "0.110603375"
      ],
      [
        "0.928779489",
        "-0.870186863",
        "0.890616132",
        "0.626831098",
        "0.336444301"
      ]
    ],
    [
      [
        "0.377189786",
        "0.581083521",
        "0.154875305",
        "-0.650431611",
        "0.749473399"
      ],
      [
        "0.074992544",
        "-0.473361793",
        "0.583602678",
        "-0.628279914",
        "0.091082028"
      ]
    ],
    [
      [
        "0.741755088",
        "-0.070418638",
        "-0.219021711",
        "-0.341479759",
        "0.842267244"
      ],
      [
        "0.260754222",
        "-0.143293704",
        "0.832810549",
        "-0.065166886",
        "0.780935502"
      ]
    ],
    [
      [
        "0.160969439",
        "0.017757996",
        "0.307675542",
        "0.987024413",
        "0.658990794"
      ],
      [
        "0.637418004",
        "-0.103351716",
        "0.032334393",
        "0.541558670",
        "0.410508652"
      ]
    ],
    [
      [
        "0.981303850",
        "0.999639033",
        "0.384843250",
        "-0.071915757",
        "0.113358058"
      ],
      [
        "0.303329738",
        "-0.249460703",
        "-0.641914229",
        "-0.952048652",
        "0.364285691"
      ]
    ]
  ]
}

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": 5,
  "k": 2
}

当前第一名的答案

{
  "subspaces": [
    [
      [
        "0.173539364",
        "0.574603856",
        "-0.651964538",
        "0.953313617",
        "0.910411373"
      ],
      [
        "0.232691397",
        "0.916200875",
        "0.881643754",
        "0.430010502",
        "0.288805831"
      ]
    ],
    [
      [
        "0.379619049",
        "0.348088699",
        "-0.759614985",
        "-0.179407142",
        "-0.700975455"
      ],
      [
        "0.598936103",
        "-0.881705793",
        "-0.697140259",
        "-0.783719015",
        "-0.836706223"
      ]
    ],
    [
      [
        "0.874987882",
        "0.355847913",
        "0.831003365",
        "-0.241211940",
        "0.541669095"
      ],
      [
        "0.666269180",
        "-0.977273782",
        "-0.289777426",
        "-0.052733496",
        "-0.275067305"
      ]
    ],
    [
      [
        "0.545821990",
        "-0.189079902",
        "0.489464943",
        "0.328822521",
        "-0.259252977"
      ],
      [
        "0.241598761",
        "-0.369473769",
        "0.551831774",
        "0.253091178",
        "-0.597409015"
      ]
    ],
    [
      [
        "0.998122567",
        "0.886485796",
        "-0.712109618",
        "0.042820452",
        "0.393869888"
      ],
      [
        "0.389488323",
        "-0.850546695",
        "-0.253187540",
        "0.220362904",
        "0.267060693"
      ]
    ],
    [
      [
        "0.430349883",
        "0.114424818",
        "0.443338735",
        "0.941938439",
        "-0.103630019"
      ],
      [
        "0.473969372",
        "-0.301521806",
        "-0.402598907",
        "0.886688677",
        "-0.930205747"
      ]
    ],
    [
      [
        "0.803305630",
        "-0.790954470",
        "-0.236301787",
        "-0.471098912",
        "0.990066196"
      ],
      [
        "0.101766642",
        "0.813838357",
        "0.842909461",
        "-0.999810445",
        "-0.984598697"
      ]
    ],
    [
      [
        "0.125168210",
        "-0.829405356",
        "0.183586775",
        "-0.139040874",
        "0.673462719"
      ],
      [
        "0.312937281",
        "-0.902046357",
        "0.834947324",
        "0.993076631",
        "-0.343519324"
      ]
    ],
    [
      [
        "0.251603899",
        "-0.423919559",
        "-0.799770895",
        "-0.136844777",
        "0.609334848"
      ],
      [
        "0.639495138",
        "-0.389327042",
        "-0.356529113",
        "0.063477340",
        "-0.334963955"
      ]
    ],
    [
      [
        "0.560630494",
        "0.245462623",
        "-0.431737592",
        "0.694510623",
        "0.396087793"
      ],
      [
        "0.594317181",
        "0.568687113",
        "-0.494881949",
        "0.678154838",
        "-0.804877113"
      ]
    ],
    [
      [
        "0.696034451",
        "0.806582008",
        "0.391594615",
        "-0.312860452",
        "-0.871534420"
      ],
      [
        "0.271879952",
        "-0.123957654",
        "0.475624188",
        "0.251958090",
        "0.462532993"
      ]
    ],
    [
      [
        "0.800400621",
        "0.036740818",
        "-0.592794343",
        "0.398029875",
        "-0.506499085"
      ],
      [
        "0.006662769",
        "-0.598961350",
        "-0.856247098",
        "0.994061820",
        "-0.291757412"
      ]
    ],
    [
      [
        "0.260399366",
        "0.380361386",
        "-0.021739362",
        "0.667227793",
        "-0.274702559"
      ],
      [
        "0.909595729",
        "-0.761655631",
        "0.488584743",
        "-0.935902257",
        "-0.175646916"
      ]
    ],
    [
      [
        "0.831089759",
        "0.514744399",
        "-0.716931845",
        "-0.993518656",
        "-0.260659596"
      ],
      [
        "0.738010883",
        "-0.096146969",
        "0.495697827",
        "-0.681582303",
        "-0.591488343"
      ]
    ],
    [
      [
        "0.106084028",
        "-0.857114161",
        "-0.069637409",
        "0.632749661",
        "-0.107472208"
      ],
      [
        "0.151990501",
        "0.067168487",
        "0.036274330",
        "-0.662307289",
        "-0.996135050"
      ]
    ],
    [
      [
        "0.342962515",
        "-0.164914051",
        "-0.947671483",
        "0.946608943",
        "0.110603375"
      ],
      [
        "0.928779489",
        "-0.870186863",
        "0.890616132",
        "0.626831098",
        "0.336444301"
      ]
    ],
    [
      [
        "0.377189786",
        "0.581083521",
        "0.154875305",
        "-0.650431611",
        "0.749473399"
      ],
      [
        "0.074992544",
        "-0.473361793",
        "0.583602678",
        "-0.628279914",
        "0.091082028"
      ]
    ],
    [
      [
        "0.741755088",
        "-0.070418638",
        "-0.219021711",
        "-0.341479759",
        "0.842267244"
      ],
      [
        "0.260754222",
        "-0.143293704",
        "0.832810549",
        "-0.065166886",
        "0.780935502"
      ]
    ],
    [
      [
        "0.160969439",
        "0.017757996",
        "0.307675542",
        "0.987024413",
        "0.658990794"
      ],
      [
        "0.637418004",
        "-0.103351716",
        "0.032334393",
        "0.541558670",
        "0.410508652"
      ]
    ],
    [
      [
        "0.981303850",
        "0.999639033",
        "0.384843250",
        "-0.071915757",
        "0.113358058"
      ],
      [
        "0.303329738",
        "-0.249460703",
        "-0.641914229",
        "-0.952048652",
        "0.364285691"
      ]
    ]
  ]
}

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

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