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 = 5, n = 15

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 = 5, n = 15

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	[0.379936775, 0.379936785]	见来源署名	github.com ↗
本站纪录	0.379936783	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15 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（11 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
0.3799367829→0.3799367828
2026-10-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.37993678296→0.37993678295
2026-09-20
StonebubuAI · Z.ai: GLM 5.3 Flash
0.379936782967→0.379936782962
2026-09-20
StonebubuAI · Z.ai: GLM 5.3 Flash
0.37993678297→0.37993678296
2026-09-19
StonebubuAI · Z.ai: GLM 5.3 Flash
0.379936782974→0.379936782971
2026-09-19
StonebubuAI · Z.ai: GLM 5.3 Flash
0.379936782977→0.379936782974
2026-09-19
StonebubuAI · Z.ai: GLM 5.3 Flash
0.37993678298→0.37993678297
2026-09-16
StonebubuAI · Z.ai: GLM 5.3 Flash
0.379936782989→0.379936782982
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.37993678299→0.37993678298
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.379936784→0.379936783
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.3799367837→0.3799367830
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": [
    [
      [
        "-0.204043955",
        "0.265211185"
      ],
      [
        "-0.349458825",
        "-0.687806364"
      ],
      [
        "0.878711900",
        "-0.276677894"
      ],
      [
        "0.999504784",
        "0.369551844"
      ],
      [
        "0.011669240",
        "0.076768676"
      ]
    ],
    [
      [
        "-0.406442576",
        "-0.958033587"
      ],
      [
        "-0.597560911",
        "0.297539874"
      ],
      [
        "-0.341158735",
        "0.736565202"
      ],
      [
        "0.023210154",
        "-0.999500588"
      ],
      [
        "0.989411049",
        "-0.382159108"
      ]
    ],
    [
      [
        "-0.999504302",
        "0.092225783"
      ],
      [
        "-0.578235353",
        "-0.105490793"
      ],
      [
        "-0.302026534",
        "-0.914529465"
      ],
      [
        "-0.478136878",
        "-0.210629532"
      ],
      [
        "0.101724900",
        "-0.542821624"
      ]
    ],
    [
      [
        "0.458351129",
        "-0.988915083"
      ],
      [
        "0.235010597",
        "0.161323614"
      ],
      [
        "0.999491328",
        "-0.141549286"
      ],
      [
        "-0.482179419",
        "0.604180383"
      ],
      [
        "-0.770605197",
        "-0.291057579"
      ]
    ],
    [
      [
        "-0.999500143",
        "0.106302717"
      ],
      [
        "0.601087046",
        "-0.146232579"
      ],
      [
        "0.129925710",
        "-0.240303697"
      ],
      [
        "-0.875247649",
        "0.060434260"
      ],
      [
        "0.367525355",
        "0.573719100"
      ]
    ],
    [
      [
        "0.512577117",
        "0.296843851"
      ],
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        "0.189997009",
        "0.299959821"
      ],
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        "-0.536833463",
        "0.177451095"
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      [
        "-0.439284544",
        "0.331157996"
      ],
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        "0.999503744",
        "0.957905424"
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      [
        "-0.360191480",
        "-0.045741592"
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        "0.999504186"
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        "0.594758470",
        "-0.715859930"
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      [
        "-0.116495659",
        "-0.049026091"
      ],
      [
        "-0.110830645",
        "0.055959098"
      ]
    ],
    [
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        "0.314339398",
        "-0.726118541"
      ],
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        "0.257712951",
        "-0.308642734"
      ],
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        "-0.466122706",
        "-0.999503915"
      ],
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        "-0.666975846",
        "-0.059539793"
      ],
      [
        "0.241922838",
        "0.491632287"
      ]
    ],
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        "0.155092136",
        "0.222642647"
      ],
      [
        "-0.623753020",
        "0.999504142"
      ],
      [
        "-0.112429876",
        "0.359866178"
      ],
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        "0.518388594",
        "-0.125538401"
      ],
      [
        "0.285924129",
        "0.464179558"
      ]
    ],
    [
      [
        "0.999500224",
        "0.373224313"
      ],
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        "-0.970870529",
        "0.428582957"
      ],
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        "0.068953468",
        "-0.659011838"
      ],
      [
        "-0.614387923",
        "0.315660025"
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        "-0.607273192",
        "-0.039392892"
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    ],
    [
      [
        "-0.249395477",
        "-0.936088436"
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      [
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        "0.327125483"
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      [
        "-0.547115045",
        "0.358458073"
      ],
      [
        "-0.674130707",
        "0.268200643"
      ],
      [
        "-0.966437129",
        "-0.670231035"
      ]
    ],
    [
      [
        "0.062018533",
        "-0.929665412"
      ],
      [
        "-0.999499122",
        "0.127346585"
      ],
      [
        "-0.035262945",
        "-0.723428800"
      ],
      [
        "0.600547790",
        "0.833887960"
      ],
      [
        "0.498729531",
        "-0.420146094"
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    ],
    [
      [
        "0.172710620",
        "0.191906583"
      ],
      [
        "-0.190650569",
        "0.311415927"
      ],
      [
        "0.442106726",
        "-0.398370417"
      ],
      [
        "-0.999497038",
        "-0.373996161"
      ],
      [
        "0.905074029",
        "-0.082320353"
      ]
    ],
    [
      [
        "0.063492261",
        "0.666895811"
      ],
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        "0.230370349",
        "0.307550770"
      ],
      [
        "0.017701653",
        "-0.403118797"
      ],
      [
        "0.214038485",
        "-0.312387055"
      ],
      [
        "0.120110096",
        "-0.999501463"
      ]
    ],
    [
      [
        "0.451054173",
        "0.290398719"
      ],
      [
        "0.039832786",
        "0.732963777"
      ],
      [
        "0.734542447",
        "-0.020700213"
      ],
      [
        "0.270639340",
        "0.999499378"
      ],
      [
        "0.284918026",
        "-0.143027197"
      ]
    ]
  ]
}

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": 15,
  "d": 5
}

当前第一名的答案

{
  "vectors": [
    [
      [
        "-0.204043955",
        "0.265211185"
      ],
      [
        "-0.349458825",
        "-0.687806364"
      ],
      [
        "0.878711900",
        "-0.276677894"
      ],
      [
        "0.999504784",
        "0.369551844"
      ],
      [
        "0.011669240",
        "0.076768676"
      ]
    ],
    [
      [
        "-0.406442576",
        "-0.958033587"
      ],
      [
        "-0.597560911",
        "0.297539874"
      ],
      [
        "-0.341158735",
        "0.736565202"
      ],
      [
        "0.023210154",
        "-0.999500588"
      ],
      [
        "0.989411049",
        "-0.382159108"
      ]
    ],
    [
      [
        "-0.999504302",
        "0.092225783"
      ],
      [
        "-0.578235353",
        "-0.105490793"
      ],
      [
        "-0.302026534",
        "-0.914529465"
      ],
      [
        "-0.478136878",
        "-0.210629532"
      ],
      [
        "0.101724900",
        "-0.542821624"
      ]
    ],
    [
      [
        "0.458351129",
        "-0.988915083"
      ],
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        "0.161323614"
      ],
      [
        "0.999491328",
        "-0.141549286"
      ],
      [
        "-0.482179419",
        "0.604180383"
      ],
      [
        "-0.770605197",
        "-0.291057579"
      ]
    ],
    [
      [
        "-0.999500143",
        "0.106302717"
      ],
      [
        "0.601087046",
        "-0.146232579"
      ],
      [
        "0.129925710",
        "-0.240303697"
      ],
      [
        "-0.875247649",
        "0.060434260"
      ],
      [
        "0.367525355",
        "0.573719100"
      ]
    ],
    [
      [
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        "0.296843851"
      ],
      [
        "0.189997009",
        "0.299959821"
      ],
      [
        "-0.536833463",
        "0.177451095"
      ],
      [
        "-0.439284544",
        "0.331157996"
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      [
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        "0.957905424"
      ]
    ],
    [
      [
        "-0.360191480",
        "-0.045741592"
      ],
      [
        "-0.289182244",
        "0.999504186"
      ],
      [
        "0.594758470",
        "-0.715859930"
      ],
      [
        "-0.116495659",
        "-0.049026091"
      ],
      [
        "-0.110830645",
        "0.055959098"
      ]
    ],
    [
      [
        "0.314339398",
        "-0.726118541"
      ],
      [
        "0.257712951",
        "-0.308642734"
      ],
      [
        "-0.466122706",
        "-0.999503915"
      ],
      [
        "-0.666975846",
        "-0.059539793"
      ],
      [
        "0.241922838",
        "0.491632287"
      ]
    ],
    [
      [
        "0.155092136",
        "0.222642647"
      ],
      [
        "-0.623753020",
        "0.999504142"
      ],
      [
        "-0.112429876",
        "0.359866178"
      ],
      [
        "0.518388594",
        "-0.125538401"
      ],
      [
        "0.285924129",
        "0.464179558"
      ]
    ],
    [
      [
        "0.999500224",
        "0.373224313"
      ],
      [
        "-0.970870529",
        "0.428582957"
      ],
      [
        "0.068953468",
        "-0.659011838"
      ],
      [
        "-0.614387923",
        "0.315660025"
      ],
      [
        "-0.607273192",
        "-0.039392892"
      ]
    ],
    [
      [
        "-0.249395477",
        "-0.936088436"
      ],
      [
        "-0.999503178",
        "0.327125483"
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      [
        "-0.547115045",
        "0.358458073"
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      [
        "-0.674130707",
        "0.268200643"
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      [
        "-0.966437129",
        "-0.670231035"
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    ],
    [
      [
        "0.062018533",
        "-0.929665412"
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      [
        "-0.999499122",
        "0.127346585"
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      [
        "-0.035262945",
        "-0.723428800"
      ],
      [
        "0.600547790",
        "0.833887960"
      ],
      [
        "0.498729531",
        "-0.420146094"
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    ],
    [
      [
        "0.172710620",
        "0.191906583"
      ],
      [
        "-0.190650569",
        "0.311415927"
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      [
        "0.442106726",
        "-0.398370417"
      ],
      [
        "-0.999497038",
        "-0.373996161"
      ],
      [
        "0.905074029",
        "-0.082320353"
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    ],
    [
      [
        "0.063492261",
        "0.666895811"
      ],
      [
        "0.230370349",
        "0.307550770"
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      [
        "0.017701653",
        "-0.403118797"
      ],
      [
        "0.214038485",
        "-0.312387055"
      ],
      [
        "0.120110096",
        "-0.999501463"
      ]
    ],
    [
      [
        "0.451054173",
        "0.290398719"
      ],
      [
        "0.039832786",
        "0.732963777"
      ],
      [
        "0.734542447",
        "-0.020700213"
      ],
      [
        "0.270639340",
        "0.999499378"
      ],
      [
        "0.284918026",
        "-0.143027197"
      ]
    ]
  ]
}

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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