P61 · 极值构型 · 经典问题 · 应用前沿 · 难

复射影空间中的码本打包 · d = 3, n = 13

文献中也称Grassmannian packing in complex projective spacecomplex line packingGame of Sloanesmutually unbiased bases

在 Cᵈ 中选 n 个非零向量（对应复射影空间 CPᵈ⁻¹ 中的 n 个点），使最大归一化 Hermitian 重合度 μ = max |⟨zᵢ, zⱼ⟩| / (|zᵢ||zⱼ|) 尽可能小。

子题d = 3, n = 13

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	[0.622143865, 0.622143875]	见来源署名	github.com ↗
本站纪录	0.622143873	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 维复空间 Cᵈ；答案是复射影空间 CPᵈ⁻¹ 中的 n 个点
提交恰好 n 个非零复向量，每个 d 个 [re, im] 对
重合度μ² 是所有复射影点对中最大的归一化 Hermitian 内积模平方；分子与分母都是有理数，验证器直接用交叉相乘精确比较 $μ^{2} = \max_{i < j} \frac{{| ⟨ z_{i}, z_{j} ⟩ |}^{2}}{{| z_{i} |}^{2} {| z_{j} |}^{2}}$
计分纪录存为 ceil(μ²·10¹⁸)；页面显示 μ，向上取整到第 9 位小数
目标在所有合法复射影码本中让 μ²（等价地 μ）尽可能小 $\min_{z_{1}, \dots, z_{n}} μ^{2}$

挑战这个纪录 ↗

当前展示：本站纪录

已验证构造13 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

复射影码本对应量子测量（如 SIC-POVM）、无线通信码本和抗噪特征表示。码字在复射影空间里分得越开，在噪声和丢包下就越容易区分。

前沿在哪里

Game of Sloanes 是一份公开维护的复射影打包榜单，收录了各参数下的当前最好结果。本页子题都选自表中已知上界与理论下界仍有明显差距的开放参数。2026 年 10 月，zzzcy 提交并经本站核验的计算机辅助证明解决了 d = 3、n = 5，确定连续最优值为 (√13 − 1)/6，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
0.6221438729→0.6221438728
2026-10-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.622143874→0.622143873
2026-09-03
StonebubuAI · Z.ai: GLM 5.3 Flash
0.622143873009→0.622143873005
2026-09-03
StonebubuAI · Z.ai: GLM 5.3 Flash
0.62214387301→0.62214387300
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.62214387302→0.62214387301
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.62214387303→0.62214387302
2026-09-01
StonebubuAI · Z.ai: GLM 5.3 Flash
0.6221438734→0.6221438730
2026-09-01

ANSWER FORMAT

答案怎么写

每个复向量是 d 个 [re, im] 对，分量写成 [-1, 1] 内的十进制字符串，最多九位小数。整体相位与非零复缩放代表同一个射影点。

提交 vectors：恰好 n 行，每行 d 个 [re, im] 对，实部虚部都是 [-1, 1] 内的十进制字符串。每行是一个非零复向量；整体相位和非零复缩放不改变答案。

当前第一名的答案

{
  "vectors": [
    [
      [
        "-0.568452811",
        "-0.069406374"
      ],
      [
        "-0.120799874",
        "0.098744024"
      ],
      [
        "-0.370442148",
        "-0.999498877"
      ]
    ],
    [
      [
        "0.999502286",
        "-0.492368232"
      ],
      [
        "-0.324758696",
        "0.285609470"
      ],
      [
        "0.511565744",
        "-0.464530034"
      ]
    ],
    [
      [
        "-0.102650494",
        "-0.388319980"
      ],
      [
        "0.053671284",
        "-0.351700635"
      ],
      [
        "-0.999499039",
        "-0.034027797"
      ]
    ],
    [
      [
        "-0.866150829",
        "0.999498896"
      ],
      [
        "0.983736180",
        "0.471507622"
      ],
      [
        "0.754573474",
        "0.730467238"
      ]
    ],
    [
      [
        "0.542563409",
        "0.042049275"
      ],
      [
        "-0.359046459",
        "0.475805780"
      ],
      [
        "-0.999500009",
        "-0.479747838"
      ]
    ],
    [
      [
        "-0.431201436",
        "-0.179147674"
      ],
      [
        "0.999500408",
        "0.168408186"
      ],
      [
        "-0.024189802",
        "0.148695499"
      ]
    ],
    [
      [
        "0.171713993",
        "0.130048785"
      ],
      [
        "0.999499436",
        "0.556675868"
      ],
      [
        "-0.948375268",
        "-0.396716779"
      ]
    ],
    [
      [
        "-0.444684139",
        "0.446196429"
      ],
      [
        "0.999499650",
        "0.095029131"
      ],
      [
        "0.065519370",
        "-0.904012459"
      ]
    ],
    [
      [
        "0.911358547",
        "0.106468621"
      ],
      [
        "0.999499483",
        "-0.390592419"
      ],
      [
        "0.224173328",
        "0.348272958"
      ]
    ],
    [
      [
        "0.169885126",
        "-0.910184480"
      ],
      [
        "0.999499088",
        "-0.334437347"
      ],
      [
        "-0.214909351",
        "-0.218657964"
      ]
    ],
    [
      [
        "0.624195014",
        "0.999499758"
      ],
      [
        "0.350990387",
        "0.274936719"
      ],
      [
        "-0.167013499",
        "-0.691617278"
      ]
    ],
    [
      [
        "-0.999500563",
        "0.294058632"
      ],
      [
        "0.167670882",
        "-0.455579772"
      ],
      [
        "0.104632505",
        "-0.509325225"
      ]
    ],
    [
      [
        "-0.125179066",
        "-0.254874984"
      ],
      [
        "0.638274899",
        "-0.999501197"
      ],
      [
        "0.435736654",
        "-0.979937609"
      ]
    ]
  ]
}

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": 13,
  "d": 3
}

当前第一名的答案

{
  "vectors": [
    [
      [
        "-0.568452811",
        "-0.069406374"
      ],
      [
        "-0.120799874",
        "0.098744024"
      ],
      [
        "-0.370442148",
        "-0.999498877"
      ]
    ],
    [
      [
        "0.999502286",
        "-0.492368232"
      ],
      [
        "-0.324758696",
        "0.285609470"
      ],
      [
        "0.511565744",
        "-0.464530034"
      ]
    ],
    [
      [
        "-0.102650494",
        "-0.388319980"
      ],
      [
        "0.053671284",
        "-0.351700635"
      ],
      [
        "-0.999499039",
        "-0.034027797"
      ]
    ],
    [
      [
        "-0.866150829",
        "0.999498896"
      ],
      [
        "0.983736180",
        "0.471507622"
      ],
      [
        "0.754573474",
        "0.730467238"
      ]
    ],
    [
      [
        "0.542563409",
        "0.042049275"
      ],
      [
        "-0.359046459",
        "0.475805780"
      ],
      [
        "-0.999500009",
        "-0.479747838"
      ]
    ],
    [
      [
        "-0.431201436",
        "-0.179147674"
      ],
      [
        "0.999500408",
        "0.168408186"
      ],
      [
        "-0.024189802",
        "0.148695499"
      ]
    ],
    [
      [
        "0.171713993",
        "0.130048785"
      ],
      [
        "0.999499436",
        "0.556675868"
      ],
      [
        "-0.948375268",
        "-0.396716779"
      ]
    ],
    [
      [
        "-0.444684139",
        "0.446196429"
      ],
      [
        "0.999499650",
        "0.095029131"
      ],
      [
        "0.065519370",
        "-0.904012459"
      ]
    ],
    [
      [
        "0.911358547",
        "0.106468621"
      ],
      [
        "0.999499483",
        "-0.390592419"
      ],
      [
        "0.224173328",
        "0.348272958"
      ]
    ],
    [
      [
        "0.169885126",
        "-0.910184480"
      ],
      [
        "0.999499088",
        "-0.334437347"
      ],
      [
        "-0.214909351",
        "-0.218657964"
      ]
    ],
    [
      [
        "0.624195014",
        "0.999499758"
      ],
      [
        "0.350990387",
        "0.274936719"
      ],
      [
        "-0.167013499",
        "-0.691617278"
      ]
    ],
    [
      [
        "-0.999500563",
        "0.294058632"
      ],
      [
        "0.167670882",
        "-0.455579772"
      ],
      [
        "0.104632505",
        "-0.509325225"
      ]
    ],
    [
      [
        "-0.125179066",
        "-0.254874984"
      ],
      [
        "0.638274899",
        "-0.999501197"
      ],
      [
        "0.435736654",
        "-0.979937609"
      ]
    ]
  ]
}

提交 vectors：恰好 n 行，每行 d 个 [re, im] 对，实部虚部都是 [-1, 1] 内的十进制字符串。每行是一个非零复向量；整体相位和非零复缩放不改变答案。 · 验证器 v1.0.0

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