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 = 14

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 = 14

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	[0.375674755, 0.375674765]	见来源署名	github.com ↗
本站纪录	0.375674759	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14 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（9 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
0.3756747585→0.3756747584
2026-10-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.375674758566→0.375674758564
2026-09-23
StonebubuAI · Z.ai: GLM 5.3 Flash
0.37567475857→0.37567475856
2026-09-20
StonebubuAI · Z.ai: GLM 5.3 Flash
0.375674758575→0.375674758571
2026-09-20
StonebubuAI · Z.ai: GLM 5.3 Flash
0.375674758577→0.375674758575
2026-09-14
StonebubuAI · Z.ai: GLM 5.3 Flash
0.37567475858→0.37567475857
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.375674758589→0.375674758581
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.3756747586→0.3756747585
2026-09-01
StonebubuAI · Z.ai: GLM 5.3 Flash
0.37567476→0.375674759
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": [
    [
      [
        "-0.265161276",
        "0.201338659"
      ],
      [
        "-0.387854670",
        "-0.139877108"
      ],
      [
        "0.538845051",
        "0.999500019"
      ],
      [
        "-0.573588113",
        "-0.408994460"
      ],
      [
        "-0.058029617",
        "0.398631280"
      ]
    ],
    [
      [
        "0.182086059",
        "-0.863854612"
      ],
      [
        "-0.334281310",
        "0.795827132"
      ],
      [
        "0.379100320",
        "-0.993301003"
      ],
      [
        "0.069413512",
        "0.124165925"
      ],
      [
        "-0.628738030",
        "0.999502577"
      ]
    ],
    [
      [
        "-0.070865576",
        "0.825757805"
      ],
      [
        "0.999503368",
        "-0.218076822"
      ],
      [
        "0.474777701",
        "-0.463804051"
      ],
      [
        "-0.154147246",
        "-0.154226976"
      ],
      [
        "0.525803430",
        "0.425526336"
      ]
    ],
    [
      [
        "0.923142499",
        "0.829614557"
      ],
      [
        "0.676099648",
        "-0.331641576"
      ],
      [
        "-0.073291816",
        "-0.451489993"
      ],
      [
        "0.490007884",
        "-0.505680490"
      ],
      [
        "-0.999499073",
        "-0.425420195"
      ]
    ],
    [
      [
        "0.440138927",
        "-0.586987674"
      ],
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        "0.796323342",
        "-0.556935609"
      ],
      [
        "0.999499915",
        "-0.628210053"
      ],
      [
        "-0.199224856",
        "-0.066597100"
      ],
      [
        "-0.314393221",
        "-0.308618330"
      ]
    ],
    [
      [
        "0.999499656",
        "-0.438060978"
      ],
      [
        "0.871132546",
        "-0.146292812"
      ],
      [
        "-0.528665160",
        "0.726633688"
      ],
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        "-0.222975780",
        "0.729075287"
      ],
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        "-0.719348337",
        "-0.834774469"
      ]
    ],
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        "0.155136387",
        "-0.280561673"
      ],
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        "0.836844539",
        "0.716684289"
      ],
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        "0.058252786",
        "0.360691457"
      ],
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        "-0.999504685",
        "0.660417801"
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        "0.400121237",
        "0.470893524"
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        "0.876724340",
        "0.859090154"
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        "0.593632012",
        "-0.534276910"
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        "-0.294391117",
        "0.999499763"
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        "-0.473065197",
        "0.277288633"
      ],
      [
        "0.007759814",
        "0.793549868"
      ]
    ],
    [
      [
        "0.998254717",
        "-0.504179881"
      ],
      [
        "0.139100742",
        "0.301291938"
      ],
      [
        "0.465144144",
        "-0.697626305"
      ],
      [
        "-0.453125644",
        "-0.829580029"
      ],
      [
        "0.999500417",
        "0.532581528"
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    ],
    [
      [
        "0.345589509",
        "-0.999500750"
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        "0.453102948",
        "0.609799574"
      ],
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        "-0.174547774",
        "-0.159195252"
      ],
      [
        "-0.108686390",
        "-0.708362897"
      ],
      [
        "-0.621917379",
        "-0.387924657"
      ]
    ],
    [
      [
        "-0.214887392",
        "0.095059756"
      ],
      [
        "0.617192012",
        "0.420405772"
      ],
      [
        "0.373107052",
        "-0.325033813"
      ],
      [
        "-0.302179562",
        "0.749439950"
      ],
      [
        "-0.999501641",
        "-0.140992404"
      ]
    ],
    [
      [
        "-0.335352620",
        "-0.691282375"
      ],
      [
        "0.383711439",
        "0.176960311"
      ],
      [
        "0.065705619",
        "0.777310025"
      ],
      [
        "0.721740628",
        "-0.164262632"
      ],
      [
        "0.212897941",
        "-0.999499844"
      ]
    ],
    [
      [
        "-0.058625843",
        "-0.119224209"
      ],
      [
        "-0.090666271",
        "0.449099663"
      ],
      [
        "0.322301343",
        "-0.019954466"
      ],
      [
        "0.999503014",
        "0.582333167"
      ],
      [
        "0.109110758",
        "0.289348636"
      ]
    ],
    [
      [
        "0.588540973",
        "-0.586441537"
      ],
      [
        "0.771132285",
        "0.582067653"
      ],
      [
        "-0.395442001",
        "0.778223693"
      ],
      [
        "0.614254507",
        "-0.210806482"
      ],
      [
        "-0.118640444",
        "0.999500244"
      ]
    ]
  ]
}

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": 14,
  "d": 5
}

当前第一名的答案

{
  "vectors": [
    [
      [
        "-0.265161276",
        "0.201338659"
      ],
      [
        "-0.387854670",
        "-0.139877108"
      ],
      [
        "0.538845051",
        "0.999500019"
      ],
      [
        "-0.573588113",
        "-0.408994460"
      ],
      [
        "-0.058029617",
        "0.398631280"
      ]
    ],
    [
      [
        "0.182086059",
        "-0.863854612"
      ],
      [
        "-0.334281310",
        "0.795827132"
      ],
      [
        "0.379100320",
        "-0.993301003"
      ],
      [
        "0.069413512",
        "0.124165925"
      ],
      [
        "-0.628738030",
        "0.999502577"
      ]
    ],
    [
      [
        "-0.070865576",
        "0.825757805"
      ],
      [
        "0.999503368",
        "-0.218076822"
      ],
      [
        "0.474777701",
        "-0.463804051"
      ],
      [
        "-0.154147246",
        "-0.154226976"
      ],
      [
        "0.525803430",
        "0.425526336"
      ]
    ],
    [
      [
        "0.923142499",
        "0.829614557"
      ],
      [
        "0.676099648",
        "-0.331641576"
      ],
      [
        "-0.073291816",
        "-0.451489993"
      ],
      [
        "0.490007884",
        "-0.505680490"
      ],
      [
        "-0.999499073",
        "-0.425420195"
      ]
    ],
    [
      [
        "0.440138927",
        "-0.586987674"
      ],
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        "0.796323342",
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        "-0.308618330"
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    [
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        "0.999499656",
        "-0.438060978"
      ],
      [
        "0.871132546",
        "-0.146292812"
      ],
      [
        "-0.528665160",
        "0.726633688"
      ],
      [
        "-0.222975780",
        "0.729075287"
      ],
      [
        "-0.719348337",
        "-0.834774469"
      ]
    ],
    [
      [
        "0.155136387",
        "-0.280561673"
      ],
      [
        "0.836844539",
        "0.716684289"
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        "0.360691457"
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        "-0.999504685",
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        "0.400121237",
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    ],
    [
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        "0.876724340",
        "0.859090154"
      ],
      [
        "0.593632012",
        "-0.534276910"
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      [
        "-0.294391117",
        "0.999499763"
      ],
      [
        "-0.473065197",
        "0.277288633"
      ],
      [
        "0.007759814",
        "0.793549868"
      ]
    ],
    [
      [
        "0.998254717",
        "-0.504179881"
      ],
      [
        "0.139100742",
        "0.301291938"
      ],
      [
        "0.465144144",
        "-0.697626305"
      ],
      [
        "-0.453125644",
        "-0.829580029"
      ],
      [
        "0.999500417",
        "0.532581528"
      ]
    ],
    [
      [
        "0.345589509",
        "-0.999500750"
      ],
      [
        "0.453102948",
        "0.609799574"
      ],
      [
        "-0.174547774",
        "-0.159195252"
      ],
      [
        "-0.108686390",
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      ],
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        "-0.621917379",
        "-0.387924657"
      ]
    ],
    [
      [
        "-0.214887392",
        "0.095059756"
      ],
      [
        "0.617192012",
        "0.420405772"
      ],
      [
        "0.373107052",
        "-0.325033813"
      ],
      [
        "-0.302179562",
        "0.749439950"
      ],
      [
        "-0.999501641",
        "-0.140992404"
      ]
    ],
    [
      [
        "-0.335352620",
        "-0.691282375"
      ],
      [
        "0.383711439",
        "0.176960311"
      ],
      [
        "0.065705619",
        "0.777310025"
      ],
      [
        "0.721740628",
        "-0.164262632"
      ],
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        "0.212897941",
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    [
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        "-0.058625843",
        "-0.119224209"
      ],
      [
        "-0.090666271",
        "0.449099663"
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        "0.322301343",
        "-0.019954466"
      ],
      [
        "0.999503014",
        "0.582333167"
      ],
      [
        "0.109110758",
        "0.289348636"
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    ],
    [
      [
        "0.588540973",
        "-0.586441537"
      ],
      [
        "0.771132285",
        "0.582067653"
      ],
      [
        "-0.395442001",
        "0.778223693"
      ],
      [
        "0.614254507",
        "-0.210806482"
      ],
      [
        "-0.118640444",
        "0.999500244"
      ]
    ]
  ]
}

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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