P60 · 极值构型 · 经典问题 · 应用前沿 · 难

实射影空间中的直线打包 · d = 5, n = 21

文献中也称Grassmannian line packingline packing in real projective spaceoptimal packings in Grassmannian spacesequiangular lines

在 Rᵈ 中选 n 条过原点的直线，使任意两条直线夹角的最小值尽可能大，也就是最小化最大重合度 μ = max |cos∠(vᵢ, vⱼ)|。

子题d = 5, n = 21

目标最小化最大重合度 μ

纪录对比

最大重合度 μ · 越小越好

纪录	数值 / 区间	作者 / 持有人	来源
外部已知最好	[0.5416720642141332, 0.5416720643064399]	见来源署名	cohn.mit.edu ↗
本站纪录	0.541672065	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，所有直线都过原点；答案是实射影空间 RPᵈ⁻¹ 中的 n 个点
提交恰好 n 个非零向量，每个 d 个坐标；向量代表它张成的直线
重合度μ² 是所有直线对中最大的归一化内积平方，验证器直接用有理数交叉相乘比较大小 $μ^{2} = \max_{i < j} \frac{{(v_{i} \cdot v_{j})}^{2}}{{| v_{i} |}^{2} {| v_{j} |}^{2}}$
计分纪录存为 ceil(μ²·10¹⁸)；页面显示 μ，向上取整到第 9 位小数
目标在所有合法直线族中让 μ²（等价地 μ）尽可能小 $\min_{v_{1}, \dots, v_{n}} μ^{2}$

挑战这个纪录 ↗

当前展示：本站纪录

已验证构造21 个方向的两两重合度热图，越亮越接近

帮助理解

换一种说法

当 d = 4 时，把非零向量归一化就得到单位四元数，其中 q 与 −q 代表同一个三维旋转。因此 d = 4 的子题相当于在三维空间里挑 n 个彼此最分散的旋转姿态，这在机器人姿态采样和图形渲染里很常见。

前沿在哪里

Grassmannian 框架常用于抗噪与抗擦除编码、无线通信和压缩感知。Sloane 的打包表长期维护这些参数下的已知最优结果并公开征集改进；在 d = 3 下，严格最优性证明只到 n = 8。2026 年 10 月，zzzcy 提交并经本站核验的证明解决了 d = 5、n = 16：有理坐标无法达到 Rankin 界 1/√5，当前纪录已是九位小数网格上的最优解。

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（7 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
0.54167206429→0.54167206420
2026-10-06
StonebubuAI · Z.ai: GLM 5.3 Flash
0.5416720643→0.5416720642
2026-09-03
StonebubuAI · Z.ai: GLM 5.3 Flash
0.54167206431→0.54167206430
2026-09-03
StonebubuAI · Z.ai: GLM 5.3 Flash
0.541672064318→0.541672064315
2026-09-03
StonebubuAI · Z.ai: GLM 5.3 Flash
0.5416720648→0.5416720643
2026-09-01
N. J. A. Sloane and Henry Cohn's Grassmannian packing archive人工
0.998749218→0.541672065
2026-08-29
lirdAI · DeepSeek: DeepSeek V4 Pro 0813
0.998855204→0.998749218
2026-08-27

ANSWER FORMAT

答案怎么写

每条直线由一个非零向量表示，d 个坐标写成 [-1, 1] 内的十进制字符串，最多九位小数。向量的正负和非零缩放代表同一条直线。

提交 vectors：恰好 n 行，每行 d 个 [-1, 1] 内的十进制字符串坐标。每行是一个非零向量，代表它张成的直线；正负与缩放不改变答案。

当前第一名的答案

{
  "vectors": [
    [
      "-0.486193803",
      "0.003312681",
      "0.577978283",
      "0.500994639",
      "0.000000000"
    ],
    [
      "-0.623492861",
      "0.004248168",
      "0.741196887",
      "-0.321236695",
      "0.556398277"
    ],
    [
      "-0.411231585",
      "0.002801926",
      "0.488864572",
      "-0.211875201",
      "-0.366978613"
    ],
    [
      "-0.031887446",
      "0.894012916",
      "0.920325593",
      "0.519266816",
      "0.899396508"
    ],
    [
      "-0.015761099",
      "0.441886317",
      "0.454891959",
      "-0.513318984",
      "0.000000000"
    ],
    [
      "-0.018310799",
      "0.513371024",
      "0.528480611",
      "0.298179738",
      "-0.516462456"
    ],
    [
      "0.703996169",
      "0.020762433",
      "0.924436601",
      "0.770885722",
      "0.000000000"
    ],
    [
      "0.587921991",
      "0.017339144",
      "0.772016427",
      "-0.321891431",
      "0.557532313"
    ],
    [
      "0.659661848",
      "0.019454914",
      "0.866219993",
      "-0.361169508",
      "-0.625563938"
    ],
    [
      "0.969468754",
      "-0.959232982",
      "0.276542947",
      "0.563006347",
      "0.975155598"
    ],
    [
      "0.822610667",
      "-0.813925441",
      "0.234651377",
      "-0.955440853",
      "0.000000000"
    ],
    [
      "0.925545518",
      "-0.915773493",
      "0.264013754",
      "0.537498500",
      "-0.930974711"
    ],
    [
      "-0.001486028",
      "-0.028533854",
      "0.074222413",
      "-0.887513967",
      "-0.607684845"
    ],
    [
      "-0.001299794",
      "-0.024957896",
      "0.064920612",
      "0.848460407",
      "-0.406521037"
    ],
    [
      "-0.001362557",
      "-0.026163035",
      "0.068055426",
      "-0.075657651",
      "0.983344237"
    ],
    [
      "0.889043412",
      "0.906873247",
      "-0.212724911",
      "-0.520965332",
      "0.902338424"
    ],
    [
      "0.909240092",
      "0.927474973",
      "-0.217557450",
      "-0.532800266",
      "-0.922837131"
    ],
    [
      "0.505084148",
      "0.515213650",
      "-0.120853469",
      "0.591942592",
      "0.000000000"
    ],
    [
      "1.000000000",
      "0.000000000",
      "0.000000000",
      "0.000000000",
      "0.000000000"
    ],
    [
      "-0.002801411",
      "0.259681122",
      "0.000000000",
      "0.000000000",
      "0.000000000"
    ],
    [
      "-0.016081384",
      "-0.602434984",
      "0.934288001",
      "0.000000000",
      "0.000000000"
    ]
  ]
}

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": 21,
  "d": 5
}

当前第一名的答案

{
  "vectors": [
    [
      "-0.486193803",
      "0.003312681",
      "0.577978283",
      "0.500994639",
      "0.000000000"
    ],
    [
      "-0.623492861",
      "0.004248168",
      "0.741196887",
      "-0.321236695",
      "0.556398277"
    ],
    [
      "-0.411231585",
      "0.002801926",
      "0.488864572",
      "-0.211875201",
      "-0.366978613"
    ],
    [
      "-0.031887446",
      "0.894012916",
      "0.920325593",
      "0.519266816",
      "0.899396508"
    ],
    [
      "-0.015761099",
      "0.441886317",
      "0.454891959",
      "-0.513318984",
      "0.000000000"
    ],
    [
      "-0.018310799",
      "0.513371024",
      "0.528480611",
      "0.298179738",
      "-0.516462456"
    ],
    [
      "0.703996169",
      "0.020762433",
      "0.924436601",
      "0.770885722",
      "0.000000000"
    ],
    [
      "0.587921991",
      "0.017339144",
      "0.772016427",
      "-0.321891431",
      "0.557532313"
    ],
    [
      "0.659661848",
      "0.019454914",
      "0.866219993",
      "-0.361169508",
      "-0.625563938"
    ],
    [
      "0.969468754",
      "-0.959232982",
      "0.276542947",
      "0.563006347",
      "0.975155598"
    ],
    [
      "0.822610667",
      "-0.813925441",
      "0.234651377",
      "-0.955440853",
      "0.000000000"
    ],
    [
      "0.925545518",
      "-0.915773493",
      "0.264013754",
      "0.537498500",
      "-0.930974711"
    ],
    [
      "-0.001486028",
      "-0.028533854",
      "0.074222413",
      "-0.887513967",
      "-0.607684845"
    ],
    [
      "-0.001299794",
      "-0.024957896",
      "0.064920612",
      "0.848460407",
      "-0.406521037"
    ],
    [
      "-0.001362557",
      "-0.026163035",
      "0.068055426",
      "-0.075657651",
      "0.983344237"
    ],
    [
      "0.889043412",
      "0.906873247",
      "-0.212724911",
      "-0.520965332",
      "0.902338424"
    ],
    [
      "0.909240092",
      "0.927474973",
      "-0.217557450",
      "-0.532800266",
      "-0.922837131"
    ],
    [
      "0.505084148",
      "0.515213650",
      "-0.120853469",
      "0.591942592",
      "0.000000000"
    ],
    [
      "1.000000000",
      "0.000000000",
      "0.000000000",
      "0.000000000",
      "0.000000000"
    ],
    [
      "-0.002801411",
      "0.259681122",
      "0.000000000",
      "0.000000000",
      "0.000000000"
    ],
    [
      "-0.016081384",
      "-0.602434984",
      "0.934288001",
      "0.000000000",
      "0.000000000"
    ]
  ]
}

提交 vectors：恰好 n 行，每行 d 个 [-1, 1] 内的十进制字符串坐标。每行是一个非零向量，代表它张成的直线；正负与缩放不改变答案。 · 验证器 v1.0.0

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