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 = 6, 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 = 6, 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	[0.327396505, 0.327396515]	见来源署名	github.com ↗
本站纪录	0.327396507	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（6 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
0.327396508→0.327396507
2026-10-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.32739650707→0.32739650706
2026-09-13
StonebubuAI · Z.ai: GLM 5.3 Flash
0.327396507076→0.327396507073
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.32739650708→0.32739650707
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.32739650709→0.32739650708
2026-09-01
StonebubuAI · Z.ai: GLM 5.3 Flash
0.3273965079→0.3273965070
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": [
    [
      [
        "0.999496457",
        "0.043502113"
      ],
      [
        "-0.182914428",
        "0.721715787"
      ],
      [
        "0.197995909",
        "0.273715070"
      ],
      [
        "0.333449075",
        "-0.587958733"
      ],
      [
        "0.790994062",
        "0.612090465"
      ],
      [
        "-0.685375934",
        "-0.252286165"
      ]
    ],
    [
      [
        "0.524824007",
        "0.372102706"
      ],
      [
        "0.054611028",
        "0.352901987"
      ],
      [
        "-0.999501272",
        "-0.568487830"
      ],
      [
        "-0.366768761",
        "-0.394935303"
      ],
      [
        "-0.555228356",
        "0.570875635"
      ],
      [
        "0.075873583",
        "-0.127692577"
      ]
    ],
    [
      [
        "0.455667232",
        "0.303726809"
      ],
      [
        "-0.998812000",
        "0.682241125"
      ],
      [
        "0.179751987",
        "-0.399855006"
      ],
      [
        "-0.558160049",
        "0.341554606"
      ],
      [
        "-0.430942313",
        "-0.140497959"
      ],
      [
        "-0.999503281",
        "-0.295839126"
      ]
    ],
    [
      [
        "-0.696709404",
        "0.159246223"
      ],
      [
        "-0.329192479",
        "-0.349684584"
      ],
      [
        "-0.413339709",
        "-0.660999333"
      ],
      [
        "0.373377429",
        "0.584491789"
      ],
      [
        "-0.144166541",
        "-0.278778422"
      ],
      [
        "-0.999500508",
        "0.181713854"
      ]
    ],
    [
      [
        "-0.327084511",
        "0.999500251"
      ],
      [
        "-0.986413803",
        "-0.851091983"
      ],
      [
        "0.264715201",
        "0.670195555"
      ],
      [
        "-0.112792903",
        "0.269783728"
      ],
      [
        "0.865688252",
        "0.535517047"
      ],
      [
        "-0.016440297",
        "0.692914981"
      ]
    ],
    [
      [
        "-0.141680186",
        "-0.577395469"
      ],
      [
        "-0.767164364",
        "-0.218694688"
      ],
      [
        "0.732005234",
        "0.212690724"
      ],
      [
        "0.945262305",
        "-0.999499975"
      ],
      [
        "0.427293352",
        "0.690448030"
      ],
      [
        "0.366708397",
        "0.458564216"
      ]
    ],
    [
      [
        "-0.145329816",
        "0.711832216"
      ],
      [
        "0.999500140",
        "-0.695362840"
      ],
      [
        "0.579587452",
        "-0.482216622"
      ],
      [
        "0.142401575",
        "0.640020009"
      ],
      [
        "0.253865257",
        "-0.073269351"
      ],
      [
        "-0.351234068",
        "0.968670317"
      ]
    ],
    [
      [
        "0.198734682",
        "0.259850847"
      ],
      [
        "-0.729789779",
        "0.078634570"
      ],
      [
        "-0.831606685",
        "-0.615605732"
      ],
      [
        "-0.197076622",
        "0.999500238"
      ],
      [
        "0.539699838",
        "0.189729407"
      ],
      [
        "0.203276186",
        "-0.219603100"
      ]
    ],
    [
      [
        "-0.301664445",
        "0.590602862"
      ],
      [
        "-0.008492121",
        "-0.187472808"
      ],
      [
        "-0.999501655",
        "0.769617988"
      ],
      [
        "-0.269211519",
        "-0.782083142"
      ],
      [
        "-0.965933223",
        "-0.686827363"
      ],
      [
        "-0.243147376",
        "-0.388609612"
      ]
    ],
    [
      [
        "-0.621634632",
        "0.061985449"
      ],
      [
        "-0.834433197",
        "0.460042455"
      ],
      [
        "0.390230955",
        "0.679829439"
      ],
      [
        "-0.520643486",
        "-0.670563502"
      ],
      [
        "0.036413536",
        "-0.662305932"
      ],
      [
        "-0.830566413",
        "0.999499661"
      ]
    ],
    [
      [
        "-0.813896586",
        "0.797217800"
      ],
      [
        "0.433286937",
        "0.297768837"
      ],
      [
        "-0.413369214",
        "0.999503221"
      ],
      [
        "-0.109724994",
        "0.483667471"
      ],
      [
        "0.300591625",
        "-0.434848821"
      ],
      [
        "0.399171488",
        "0.316482434"
      ]
    ],
    [
      [
        "-0.999501791",
        "-0.145608086"
      ],
      [
        "-0.383847991",
        "0.061034040"
      ],
      [
        "0.082406329",
        "0.128254539"
      ],
      [
        "0.132484976",
        "0.366670676"
      ],
      [
        "0.762595729",
        "0.739736412"
      ],
      [
        "0.367807992",
        "-0.807356179"
      ]
    ],
    [
      [
        "-0.909735373",
        "0.249765507"
      ],
      [
        "0.349255777",
        "0.915075238"
      ],
      [
        "0.037085373",
        "0.099432992"
      ],
      [
        "0.590734016",
        "-0.476792713"
      ],
      [
        "-0.495866668",
        "0.100879849"
      ],
      [
        "0.999500060",
        "-0.487354050"
      ]
    ],
    [
      [
        "0.103741122",
        "-0.063733071"
      ],
      [
        "-0.420296475",
        "-0.304534879"
      ],
      [
        "-0.351765066",
        "0.400347438"
      ],
      [
        "-0.999501370",
        "-0.130160158"
      ],
      [
        "0.167088260",
        "-0.273917624"
      ],
      [
        "0.475273810",
        "0.274890117"
      ]
    ],
    [
      [
        "0.423946944",
        "-0.586239333"
      ],
      [
        "-0.060479576",
        "-0.548452690"
      ],
      [
        "-0.025742745",
        "0.343179159"
      ],
      [
        "0.069447081",
        "0.162964996"
      ],
      [
        "-0.296984275",
        "-0.999499550"
      ],
      [
        "0.000718866",
        "-0.420294697"
      ]
    ]
  ]
}

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": 6
}

当前第一名的答案

{
  "vectors": [
    [
      [
        "0.999496457",
        "0.043502113"
      ],
      [
        "-0.182914428",
        "0.721715787"
      ],
      [
        "0.197995909",
        "0.273715070"
      ],
      [
        "0.333449075",
        "-0.587958733"
      ],
      [
        "0.790994062",
        "0.612090465"
      ],
      [
        "-0.685375934",
        "-0.252286165"
      ]
    ],
    [
      [
        "0.524824007",
        "0.372102706"
      ],
      [
        "0.054611028",
        "0.352901987"
      ],
      [
        "-0.999501272",
        "-0.568487830"
      ],
      [
        "-0.366768761",
        "-0.394935303"
      ],
      [
        "-0.555228356",
        "0.570875635"
      ],
      [
        "0.075873583",
        "-0.127692577"
      ]
    ],
    [
      [
        "0.455667232",
        "0.303726809"
      ],
      [
        "-0.998812000",
        "0.682241125"
      ],
      [
        "0.179751987",
        "-0.399855006"
      ],
      [
        "-0.558160049",
        "0.341554606"
      ],
      [
        "-0.430942313",
        "-0.140497959"
      ],
      [
        "-0.999503281",
        "-0.295839126"
      ]
    ],
    [
      [
        "-0.696709404",
        "0.159246223"
      ],
      [
        "-0.329192479",
        "-0.349684584"
      ],
      [
        "-0.413339709",
        "-0.660999333"
      ],
      [
        "0.373377429",
        "0.584491789"
      ],
      [
        "-0.144166541",
        "-0.278778422"
      ],
      [
        "-0.999500508",
        "0.181713854"
      ]
    ],
    [
      [
        "-0.327084511",
        "0.999500251"
      ],
      [
        "-0.986413803",
        "-0.851091983"
      ],
      [
        "0.264715201",
        "0.670195555"
      ],
      [
        "-0.112792903",
        "0.269783728"
      ],
      [
        "0.865688252",
        "0.535517047"
      ],
      [
        "-0.016440297",
        "0.692914981"
      ]
    ],
    [
      [
        "-0.141680186",
        "-0.577395469"
      ],
      [
        "-0.767164364",
        "-0.218694688"
      ],
      [
        "0.732005234",
        "0.212690724"
      ],
      [
        "0.945262305",
        "-0.999499975"
      ],
      [
        "0.427293352",
        "0.690448030"
      ],
      [
        "0.366708397",
        "0.458564216"
      ]
    ],
    [
      [
        "-0.145329816",
        "0.711832216"
      ],
      [
        "0.999500140",
        "-0.695362840"
      ],
      [
        "0.579587452",
        "-0.482216622"
      ],
      [
        "0.142401575",
        "0.640020009"
      ],
      [
        "0.253865257",
        "-0.073269351"
      ],
      [
        "-0.351234068",
        "0.968670317"
      ]
    ],
    [
      [
        "0.198734682",
        "0.259850847"
      ],
      [
        "-0.729789779",
        "0.078634570"
      ],
      [
        "-0.831606685",
        "-0.615605732"
      ],
      [
        "-0.197076622",
        "0.999500238"
      ],
      [
        "0.539699838",
        "0.189729407"
      ],
      [
        "0.203276186",
        "-0.219603100"
      ]
    ],
    [
      [
        "-0.301664445",
        "0.590602862"
      ],
      [
        "-0.008492121",
        "-0.187472808"
      ],
      [
        "-0.999501655",
        "0.769617988"
      ],
      [
        "-0.269211519",
        "-0.782083142"
      ],
      [
        "-0.965933223",
        "-0.686827363"
      ],
      [
        "-0.243147376",
        "-0.388609612"
      ]
    ],
    [
      [
        "-0.621634632",
        "0.061985449"
      ],
      [
        "-0.834433197",
        "0.460042455"
      ],
      [
        "0.390230955",
        "0.679829439"
      ],
      [
        "-0.520643486",
        "-0.670563502"
      ],
      [
        "0.036413536",
        "-0.662305932"
      ],
      [
        "-0.830566413",
        "0.999499661"
      ]
    ],
    [
      [
        "-0.813896586",
        "0.797217800"
      ],
      [
        "0.433286937",
        "0.297768837"
      ],
      [
        "-0.413369214",
        "0.999503221"
      ],
      [
        "-0.109724994",
        "0.483667471"
      ],
      [
        "0.300591625",
        "-0.434848821"
      ],
      [
        "0.399171488",
        "0.316482434"
      ]
    ],
    [
      [
        "-0.999501791",
        "-0.145608086"
      ],
      [
        "-0.383847991",
        "0.061034040"
      ],
      [
        "0.082406329",
        "0.128254539"
      ],
      [
        "0.132484976",
        "0.366670676"
      ],
      [
        "0.762595729",
        "0.739736412"
      ],
      [
        "0.367807992",
        "-0.807356179"
      ]
    ],
    [
      [
        "-0.909735373",
        "0.249765507"
      ],
      [
        "0.349255777",
        "0.915075238"
      ],
      [
        "0.037085373",
        "0.099432992"
      ],
      [
        "0.590734016",
        "-0.476792713"
      ],
      [
        "-0.495866668",
        "0.100879849"
      ],
      [
        "0.999500060",
        "-0.487354050"
      ]
    ],
    [
      [
        "0.103741122",
        "-0.063733071"
      ],
      [
        "-0.420296475",
        "-0.304534879"
      ],
      [
        "-0.351765066",
        "0.400347438"
      ],
      [
        "-0.999501370",
        "-0.130160158"
      ],
      [
        "0.167088260",
        "-0.273917624"
      ],
      [
        "0.475273810",
        "0.274890117"
      ]
    ],
    [
      [
        "0.423946944",
        "-0.586239333"
      ],
      [
        "-0.060479576",
        "-0.548452690"
      ],
      [
        "-0.025742745",
        "0.343179159"
      ],
      [
        "0.069447081",
        "0.162964996"
      ],
      [
        "-0.296984275",
        "-0.999499550"
      ],
      [
        "0.000718866",
        "-0.420294697"
      ]
    ]
  ]
}

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

DISCUSSION

讨论区

欢迎交流解题思路、分享构造方法或探讨卡壳之处。所有登录用户均可发帖；发言使用与排行榜一致的公开署名，并附带唯一注册编号（#UID）以防冒名。新发言经审核后公开。

暂无讨论，欢迎发表第一篇帖子。