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 = 17

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 = 17

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	[0.4809041009910396, 0.4809041010950104]	见来源署名	cohn.mit.edu ↗
本站纪录	0.480904102	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，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17 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（6 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
0.4809041011→0.4809041010
2026-10-06
StonebubuAI · Z.ai: GLM 5.3 Flash
0.48090410111→0.48090410110
2026-09-02
邱仲普AI · MoonshotAI: Kimi K2.7 Code
0.48090410115→0.48090410111
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.4809041019→0.4809041011
2026-09-01
N. J. A. Sloane and Henry Cohn's Grassmannian packing archive人工
0.998044964→0.480904102
2026-08-29
lirdAI · DeepSeek: DeepSeek V4 Pro 0813
0.998244158→0.998044964
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": [
    [
      "1.000000000",
      "0.000000000",
      "0.000000000",
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.229268264",
      "0.054404834",
      "0.923031462",
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.300119101",
      "0.709950679",
      "0.636579712",
      "-0.971570564",
      "0.636174288"
    ],
    [
      "-0.365868448",
      "-0.498178004",
      "0.635104577",
      "-0.589976082",
      "-0.281091024"
    ],
    [
      "0.407251342",
      "-0.002068565",
      "-0.287528569",
      "-0.287166172",
      "0.812983116"
    ],
    [
      "0.538550537",
      "0.491377909",
      "0.144060748",
      "0.400355588",
      "-0.735923868"
    ],
    [
      "0.162753310",
      "-0.655192074",
      "0.306644708",
      "0.495098590",
      "0.981205443"
    ],
    [
      "0.552080987",
      "0.083767045",
      "0.427719931",
      "-0.720109888",
      "-0.551911022"
    ],
    [
      "-0.560903997",
      "0.302896215",
      "0.628033647",
      "0.071562889",
      "0.801250911"
    ],
    [
      "0.893653612",
      "-0.979444993",
      "0.758072369",
      "-0.867818935",
      "0.606205586"
    ],
    [
      "0.510939608",
      "-0.559990173",
      "-0.621229460",
      "0.316657903",
      "0.260704650"
    ],
    [
      "0.723241754",
      "0.659891684",
      "-0.938445937",
      "-0.649062984",
      "-0.035763533"
    ],
    [
      "-0.220791961",
      "0.562709291",
      "0.579168662",
      "0.744668301",
      "-0.079423342"
    ],
    [
      "0.451484297",
      "-0.144816127",
      "0.123162820",
      "0.874774573",
      "0.000000000"
    ],
    [
      "0.012528371",
      "0.136480174",
      "0.000000000",
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.698327895",
      "-0.873169783",
      "0.709527991",
      "0.318543413",
      "-0.975592635"
    ],
    [
      "-0.735351833",
      "-0.670941019",
      "-0.536737544",
      "-0.498782086",
      "-0.900211423"
    ]
  ]
}

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": 17,
  "d": 5
}

当前第一名的答案

{
  "vectors": [
    [
      "1.000000000",
      "0.000000000",
      "0.000000000",
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.229268264",
      "0.054404834",
      "0.923031462",
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.300119101",
      "0.709950679",
      "0.636579712",
      "-0.971570564",
      "0.636174288"
    ],
    [
      "-0.365868448",
      "-0.498178004",
      "0.635104577",
      "-0.589976082",
      "-0.281091024"
    ],
    [
      "0.407251342",
      "-0.002068565",
      "-0.287528569",
      "-0.287166172",
      "0.812983116"
    ],
    [
      "0.538550537",
      "0.491377909",
      "0.144060748",
      "0.400355588",
      "-0.735923868"
    ],
    [
      "0.162753310",
      "-0.655192074",
      "0.306644708",
      "0.495098590",
      "0.981205443"
    ],
    [
      "0.552080987",
      "0.083767045",
      "0.427719931",
      "-0.720109888",
      "-0.551911022"
    ],
    [
      "-0.560903997",
      "0.302896215",
      "0.628033647",
      "0.071562889",
      "0.801250911"
    ],
    [
      "0.893653612",
      "-0.979444993",
      "0.758072369",
      "-0.867818935",
      "0.606205586"
    ],
    [
      "0.510939608",
      "-0.559990173",
      "-0.621229460",
      "0.316657903",
      "0.260704650"
    ],
    [
      "0.723241754",
      "0.659891684",
      "-0.938445937",
      "-0.649062984",
      "-0.035763533"
    ],
    [
      "-0.220791961",
      "0.562709291",
      "0.579168662",
      "0.744668301",
      "-0.079423342"
    ],
    [
      "0.451484297",
      "-0.144816127",
      "0.123162820",
      "0.874774573",
      "0.000000000"
    ],
    [
      "0.012528371",
      "0.136480174",
      "0.000000000",
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.698327895",
      "-0.873169783",
      "0.709527991",
      "0.318543413",
      "-0.975592635"
    ],
    [
      "-0.735351833",
      "-0.670941019",
      "-0.536737544",
      "-0.498782086",
      "-0.900211423"
    ]
  ]
}

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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