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 = 18

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 = 18

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	[0.4839990002572319, 0.4839990003605379]	见来源署名	cohn.mit.edu ↗
本站纪录	0.483999001	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18 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
0.48399900025→0.48399900023
2026-10-06
zzzcyAI · OpenAI: GPT-6 Astra
0.4839990003→0.4839990002
2026-10-04
StonebubuAI · Z.ai: GLM 5.3 Flash
0.48399900034→0.48399900033
2026-09-04
StonebubuAI · Z.ai: GLM 5.3 Flash
0.483999000348→0.483999000342
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.4839990008→0.4839990003
2026-09-01
N. J. A. Sloane and Henry Cohn's Grassmannian packing archive人工
0.998268397→0.483999001
2026-08-29
lirdAI · DeepSeek: DeepSeek V4 Pro 0813
0.998436338→0.998268397
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": [
    [
      "0.253205316",
      "-0.253205316",
      "0.226525221",
      "0.306837190",
      "0.000000000"
    ],
    [
      "0.941472246",
      "-0.941472246",
      "0.842269870",
      "-0.570443589",
      "0.988037279"
    ],
    [
      "0.808007383",
      "-0.808007383",
      "0.722868121",
      "-0.489576441",
      "-0.847971270"
    ],
    [
      "0.822778463",
      "0.822778463",
      "0.736082781",
      "0.008079608",
      "-0.997019914"
    ],
    [
      "0.939132269",
      "0.939132269",
      "0.840176455",
      "0.980938111",
      "0.576993752"
    ],
    [
      "0.726910517",
      "0.726910517",
      "0.650316384",
      "-0.766407418",
      "0.434243053"
    ],
    [
      "0.000000000",
      "0.000000000",
      "0.087585163",
      "-0.489510785",
      "-0.133292138"
    ],
    [
      "0.000000000",
      "0.000000000",
      "0.093445952",
      "0.384291983",
      "-0.381190392"
    ],
    [
      "0.000000000",
      "0.000000000",
      "0.133338920",
      "0.196877232",
      "0.746847042"
    ],
    [
      "-0.726910517",
      "-0.726910517",
      "0.650316384",
      "-0.766407418",
      "0.434243053"
    ],
    [
      "-0.822778463",
      "-0.822778463",
      "0.736082781",
      "0.008079608",
      "-0.997019914"
    ],
    [
      "-0.939132269",
      "-0.939132269",
      "0.840176455",
      "0.980938111",
      "0.576993752"
    ],
    [
      "-0.941472246",
      "0.941472246",
      "0.842269870",
      "-0.570443589",
      "0.988037279"
    ],
    [
      "-0.808007383",
      "0.808007383",
      "0.722868121",
      "-0.489576441",
      "-0.847971270"
    ],
    [
      "-0.253205316",
      "0.253205316",
      "0.226525221",
      "0.306837190",
      "0.000000000"
    ],
    [
      "1.000000000",
      "0.000000000",
      "0.000000000",
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.000000000",
      "1.000000000",
      "0.000000000",
      "0.000000000",
      "0.000000000"
    ],
    [
      "-0.026526344",
      "-0.032655716",
      "0.635059891",
      "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": 18,
  "d": 5
}

当前第一名的答案

{
  "vectors": [
    [
      "0.253205316",
      "-0.253205316",
      "0.226525221",
      "0.306837190",
      "0.000000000"
    ],
    [
      "0.941472246",
      "-0.941472246",
      "0.842269870",
      "-0.570443589",
      "0.988037279"
    ],
    [
      "0.808007383",
      "-0.808007383",
      "0.722868121",
      "-0.489576441",
      "-0.847971270"
    ],
    [
      "0.822778463",
      "0.822778463",
      "0.736082781",
      "0.008079608",
      "-0.997019914"
    ],
    [
      "0.939132269",
      "0.939132269",
      "0.840176455",
      "0.980938111",
      "0.576993752"
    ],
    [
      "0.726910517",
      "0.726910517",
      "0.650316384",
      "-0.766407418",
      "0.434243053"
    ],
    [
      "0.000000000",
      "0.000000000",
      "0.087585163",
      "-0.489510785",
      "-0.133292138"
    ],
    [
      "0.000000000",
      "0.000000000",
      "0.093445952",
      "0.384291983",
      "-0.381190392"
    ],
    [
      "0.000000000",
      "0.000000000",
      "0.133338920",
      "0.196877232",
      "0.746847042"
    ],
    [
      "-0.726910517",
      "-0.726910517",
      "0.650316384",
      "-0.766407418",
      "0.434243053"
    ],
    [
      "-0.822778463",
      "-0.822778463",
      "0.736082781",
      "0.008079608",
      "-0.997019914"
    ],
    [
      "-0.939132269",
      "-0.939132269",
      "0.840176455",
      "0.980938111",
      "0.576993752"
    ],
    [
      "-0.941472246",
      "0.941472246",
      "0.842269870",
      "-0.570443589",
      "0.988037279"
    ],
    [
      "-0.808007383",
      "0.808007383",
      "0.722868121",
      "-0.489576441",
      "-0.847971270"
    ],
    [
      "-0.253205316",
      "0.253205316",
      "0.226525221",
      "0.306837190",
      "0.000000000"
    ],
    [
      "1.000000000",
      "0.000000000",
      "0.000000000",
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.000000000",
      "1.000000000",
      "0.000000000",
      "0.000000000",
      "0.000000000"
    ],
    [
      "-0.026526344",
      "-0.032655716",
      "0.635059891",
      "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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