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 = 16

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 = 16

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	[0.388092835, 0.388092845]	见来源署名	github.com ↗
本站纪录	0.388092844	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16 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（8 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
0.3880928434→0.3880928433
2026-10-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.388092843419→0.388092843418
2026-09-08
StonebubuAI · Z.ai: GLM 5.3 Flash
0.38809284342→0.38809284341
2026-09-03
StonebubuAI · Z.ai: GLM 5.3 Flash
0.388092843426→0.388092843423
2026-09-03
StonebubuAI · Z.ai: GLM 5.3 Flash
0.38809284343→0.38809284342
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.388092843437→0.388092843433
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.38809284345→0.38809284343
2026-09-01
StonebubuAI · Z.ai: GLM 5.3 Flash
0.388092845→0.388092844
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": [
    [
      [
        "0.804530131",
        "0.177488590"
      ],
      [
        "-0.465191723",
        "-0.999506863"
      ],
      [
        "-0.899527906",
        "-0.406321349"
      ],
      [
        "0.906798153",
        "0.030984542"
      ],
      [
        "0.464754879",
        "0.575376753"
      ]
    ],
    [
      [
        "-0.999498110",
        "0.043369979"
      ],
      [
        "-0.211293961",
        "-0.460842397"
      ],
      [
        "-0.751098797",
        "0.080951707"
      ],
      [
        "0.413331206",
        "-0.338696001"
      ],
      [
        "0.338804852",
        "0.253205410"
      ]
    ],
    [
      [
        "0.999503546",
        "-0.300208868"
      ],
      [
        "0.127226705",
        "-0.385385848"
      ],
      [
        "-0.000155323",
        "0.513387496"
      ],
      [
        "0.277027230",
        "0.658388422"
      ],
      [
        "0.663818154",
        "-0.480487377"
      ]
    ],
    [
      [
        "0.490562247",
        "0.135096061"
      ],
      [
        "0.999495128",
        "-0.140304641"
      ],
      [
        "0.297641968",
        "0.671637479"
      ],
      [
        "0.014290753",
        "0.144566648"
      ],
      [
        "-0.321535979",
        "0.408738573"
      ]
    ],
    [
      [
        "-0.122740656",
        "-0.201883316"
      ],
      [
        "0.999501731",
        "0.229715274"
      ],
      [
        "-0.836717041",
        "-0.720539794"
      ],
      [
        "-0.761522472",
        "0.131291330"
      ],
      [
        "-0.164927061",
        "-0.165179835"
      ]
    ],
    [
      [
        "0.058978117",
        "-0.037870024"
      ],
      [
        "0.404833903",
        "0.794017631"
      ],
      [
        "0.254657917",
        "-0.208880182"
      ],
      [
        "0.493032467",
        "0.312843069"
      ],
      [
        "0.018003014",
        "-0.999505885"
      ]
    ],
    [
      [
        "-0.312770560",
        "0.999497200"
      ],
      [
        "0.561667391",
        "0.055306889"
      ],
      [
        "-0.717656920",
        "-0.744499150"
      ],
      [
        "0.473796699",
        "-0.279372133"
      ],
      [
        "0.350595169",
        "-0.445861219"
      ]
    ],
    [
      [
        "0.046697310",
        "0.032404547"
      ],
      [
        "0.538107507",
        "0.534516103"
      ],
      [
        "0.682910626",
        "-0.386359249"
      ],
      [
        "0.720131732",
        "-0.303161170"
      ],
      [
        "-0.483339400",
        "0.999499393"
      ]
    ],
    [
      [
        "0.958548719",
        "0.723983966"
      ],
      [
        "-0.999495546",
        "-0.022681805"
      ],
      [
        "-0.377602953",
        "-0.793544502"
      ],
      [
        "-0.736170608",
        "-0.803936742"
      ],
      [
        "0.151711192",
        "0.150949346"
      ]
    ],
    [
      [
        "0.600355508",
        "0.067462980"
      ],
      [
        "-0.002679947",
        "-0.219796755"
      ],
      [
        "-0.096594216",
        "-0.999500058"
      ],
      [
        "-0.036280917",
        "0.337100146"
      ],
      [
        "0.348145640",
        "-0.698671793"
      ]
    ],
    [
      [
        "0.154078589",
        "-0.090956319"
      ],
      [
        "-0.002602365",
        "0.215654767"
      ],
      [
        "-0.269693506",
        "-0.041413668"
      ],
      [
        "-0.574512038",
        "0.529981929"
      ],
      [
        "-0.999500839",
        "0.680801902"
      ]
    ],
    [
      [
        "0.988928239",
        "-0.683080053"
      ],
      [
        "0.359045990",
        "0.630326091"
      ],
      [
        "0.541385554",
        "0.477730869"
      ],
      [
        "-0.253235612",
        "-0.501300837"
      ],
      [
        "0.999499901",
        "0.598521374"
      ]
    ],
    [
      [
        "-0.999501195",
        "-0.254078374"
      ],
      [
        "-0.288394170",
        "0.144245470"
      ],
      [
        "-0.368899306",
        "0.211136453"
      ],
      [
        "-0.701184576",
        "0.486423272"
      ],
      [
        "-0.043217249",
        "0.179294144"
      ]
    ],
    [
      [
        "-0.169222947",
        "0.163772152"
      ],
      [
        "-0.069245020",
        "0.535875321"
      ],
      [
        "-0.234356274",
        "0.143406988"
      ],
      [
        "0.999493779",
        "-0.729515162"
      ],
      [
        "-0.081828856",
        "-0.096359884"
      ]
    ],
    [
      [
        "-0.895533428",
        "-0.119569353"
      ],
      [
        "0.853099131",
        "0.936547769"
      ],
      [
        "-0.567584083",
        "0.395699700"
      ],
      [
        "-0.283177624",
        "-0.400356056"
      ],
      [
        "0.999498762",
        "0.280283552"
      ]
    ],
    [
      [
        "0.669105178",
        "-0.422865031"
      ],
      [
        "0.999498556",
        "0.357439561"
      ],
      [
        "-0.936170736",
        "-0.002562825"
      ],
      [
        "0.809807088",
        "-0.061988172"
      ],
      [
        "0.330631870",
        "0.165509533"
      ]
    ]
  ]
}

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": 16,
  "d": 5
}

当前第一名的答案

{
  "vectors": [
    [
      [
        "0.804530131",
        "0.177488590"
      ],
      [
        "-0.465191723",
        "-0.999506863"
      ],
      [
        "-0.899527906",
        "-0.406321349"
      ],
      [
        "0.906798153",
        "0.030984542"
      ],
      [
        "0.464754879",
        "0.575376753"
      ]
    ],
    [
      [
        "-0.999498110",
        "0.043369979"
      ],
      [
        "-0.211293961",
        "-0.460842397"
      ],
      [
        "-0.751098797",
        "0.080951707"
      ],
      [
        "0.413331206",
        "-0.338696001"
      ],
      [
        "0.338804852",
        "0.253205410"
      ]
    ],
    [
      [
        "0.999503546",
        "-0.300208868"
      ],
      [
        "0.127226705",
        "-0.385385848"
      ],
      [
        "-0.000155323",
        "0.513387496"
      ],
      [
        "0.277027230",
        "0.658388422"
      ],
      [
        "0.663818154",
        "-0.480487377"
      ]
    ],
    [
      [
        "0.490562247",
        "0.135096061"
      ],
      [
        "0.999495128",
        "-0.140304641"
      ],
      [
        "0.297641968",
        "0.671637479"
      ],
      [
        "0.014290753",
        "0.144566648"
      ],
      [
        "-0.321535979",
        "0.408738573"
      ]
    ],
    [
      [
        "-0.122740656",
        "-0.201883316"
      ],
      [
        "0.999501731",
        "0.229715274"
      ],
      [
        "-0.836717041",
        "-0.720539794"
      ],
      [
        "-0.761522472",
        "0.131291330"
      ],
      [
        "-0.164927061",
        "-0.165179835"
      ]
    ],
    [
      [
        "0.058978117",
        "-0.037870024"
      ],
      [
        "0.404833903",
        "0.794017631"
      ],
      [
        "0.254657917",
        "-0.208880182"
      ],
      [
        "0.493032467",
        "0.312843069"
      ],
      [
        "0.018003014",
        "-0.999505885"
      ]
    ],
    [
      [
        "-0.312770560",
        "0.999497200"
      ],
      [
        "0.561667391",
        "0.055306889"
      ],
      [
        "-0.717656920",
        "-0.744499150"
      ],
      [
        "0.473796699",
        "-0.279372133"
      ],
      [
        "0.350595169",
        "-0.445861219"
      ]
    ],
    [
      [
        "0.046697310",
        "0.032404547"
      ],
      [
        "0.538107507",
        "0.534516103"
      ],
      [
        "0.682910626",
        "-0.386359249"
      ],
      [
        "0.720131732",
        "-0.303161170"
      ],
      [
        "-0.483339400",
        "0.999499393"
      ]
    ],
    [
      [
        "0.958548719",
        "0.723983966"
      ],
      [
        "-0.999495546",
        "-0.022681805"
      ],
      [
        "-0.377602953",
        "-0.793544502"
      ],
      [
        "-0.736170608",
        "-0.803936742"
      ],
      [
        "0.151711192",
        "0.150949346"
      ]
    ],
    [
      [
        "0.600355508",
        "0.067462980"
      ],
      [
        "-0.002679947",
        "-0.219796755"
      ],
      [
        "-0.096594216",
        "-0.999500058"
      ],
      [
        "-0.036280917",
        "0.337100146"
      ],
      [
        "0.348145640",
        "-0.698671793"
      ]
    ],
    [
      [
        "0.154078589",
        "-0.090956319"
      ],
      [
        "-0.002602365",
        "0.215654767"
      ],
      [
        "-0.269693506",
        "-0.041413668"
      ],
      [
        "-0.574512038",
        "0.529981929"
      ],
      [
        "-0.999500839",
        "0.680801902"
      ]
    ],
    [
      [
        "0.988928239",
        "-0.683080053"
      ],
      [
        "0.359045990",
        "0.630326091"
      ],
      [
        "0.541385554",
        "0.477730869"
      ],
      [
        "-0.253235612",
        "-0.501300837"
      ],
      [
        "0.999499901",
        "0.598521374"
      ]
    ],
    [
      [
        "-0.999501195",
        "-0.254078374"
      ],
      [
        "-0.288394170",
        "0.144245470"
      ],
      [
        "-0.368899306",
        "0.211136453"
      ],
      [
        "-0.701184576",
        "0.486423272"
      ],
      [
        "-0.043217249",
        "0.179294144"
      ]
    ],
    [
      [
        "-0.169222947",
        "0.163772152"
      ],
      [
        "-0.069245020",
        "0.535875321"
      ],
      [
        "-0.234356274",
        "0.143406988"
      ],
      [
        "0.999493779",
        "-0.729515162"
      ],
      [
        "-0.081828856",
        "-0.096359884"
      ]
    ],
    [
      [
        "-0.895533428",
        "-0.119569353"
      ],
      [
        "0.853099131",
        "0.936547769"
      ],
      [
        "-0.567584083",
        "0.395699700"
      ],
      [
        "-0.283177624",
        "-0.400356056"
      ],
      [
        "0.999498762",
        "0.280283552"
      ]
    ],
    [
      [
        "0.669105178",
        "-0.422865031"
      ],
      [
        "0.999498556",
        "0.357439561"
      ],
      [
        "-0.936170736",
        "-0.002562825"
      ],
      [
        "0.809807088",
        "-0.061988172"
      ],
      [
        "0.330631870",
        "0.165509533"
      ]
    ]
  ]
}

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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