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 = 10

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 = 10

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	[0.272203145, 0.272203155]	见来源署名	github.com ↗
本站纪录	0.27220315	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10 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
0.2722031499→0.2722031498
2026-10-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.272203149948→0.272203149945
2026-09-06
StonebubuAI · Z.ai: GLM 5.3 Flash
0.27220314995→0.27220314994
2026-09-03
StonebubuAI · Z.ai: GLM 5.3 Flash
0.272203149958→0.272203149952
2026-09-03
StonebubuAI · Z.ai: GLM 5.3 Flash
0.27220314996→0.27220314995
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.272203149967→0.272203149962
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.272203152→0.27220315
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": [
    [
      [
        "0.999499563",
        "-0.199572763"
      ],
      [
        "0.447251077",
        "0.544147372"
      ],
      [
        "-0.201225674",
        "-0.200923148"
      ],
      [
        "0.827632643",
        "-0.009505971"
      ],
      [
        "0.183630022",
        "-0.590664826"
      ],
      [
        "-0.857751470",
        "0.840445603"
      ]
    ],
    [
      [
        "-0.631551936",
        "0.576236980"
      ],
      [
        "0.157957888",
        "-0.889323906"
      ],
      [
        "-0.064480125",
        "-0.457339132"
      ],
      [
        "0.312570108",
        "-0.219637271"
      ],
      [
        "0.999498778",
        "-0.434655746"
      ],
      [
        "-0.184185466",
        "-0.379671753"
      ]
    ],
    [
      [
        "-0.107739484",
        "-0.372929224"
      ],
      [
        "-0.682380717",
        "-0.634242056"
      ],
      [
        "-0.022807240",
        "0.299100304"
      ],
      [
        "0.522167562",
        "0.035576823"
      ],
      [
        "0.755496832",
        "0.489289283"
      ],
      [
        "-0.999500321",
        "0.408357269"
      ]
    ],
    [
      [
        "0.085609340",
        "0.334557121"
      ],
      [
        "0.212163855",
        "-0.032144926"
      ],
      [
        "-0.419079037",
        "0.999500149"
      ],
      [
        "-0.916167037",
        "0.638067645"
      ],
      [
        "-0.061269982",
        "0.041415189"
      ],
      [
        "-0.177171120",
        "-0.011015412"
      ]
    ],
    [
      [
        "0.475662476",
        "-0.129303141"
      ],
      [
        "-0.288035533",
        "0.578535209"
      ],
      [
        "0.150512997",
        "0.005309555"
      ],
      [
        "-0.029899101",
        "-0.781459188"
      ],
      [
        "0.302919024",
        "0.999499880"
      ],
      [
        "-0.415374760",
        "-0.532983748"
      ]
    ],
    [
      [
        "-0.465632395",
        "-0.321423840"
      ],
      [
        "-0.860847307",
        "0.868039111"
      ],
      [
        "-0.137602902",
        "0.020983683"
      ],
      [
        "0.486166011",
        "0.999500208"
      ],
      [
        "-0.096529674",
        "-0.234682355"
      ],
      [
        "0.260624277",
        "-0.075591580"
      ]
    ],
    [
      [
        "-0.399270392",
        "0.171133064"
      ],
      [
        "-0.521923524",
        "0.148353816"
      ],
      [
        "-0.485164438",
        "0.999500208"
      ],
      [
        "0.326058948",
        "-0.534183683"
      ],
      [
        "0.435506251",
        "-0.203446001"
      ],
      [
        "0.240046744",
        "-0.556421607"
      ]
    ],
    [
      [
        "-0.655431546",
        "-0.999500928"
      ],
      [
        "0.292935242",
        "0.246177527"
      ],
      [
        "-0.503815266",
        "0.713537922"
      ],
      [
        "0.086975686",
        "0.077165556"
      ],
      [
        "-0.219811237",
        "-0.164700604"
      ],
      [
        "-0.667966908",
        "-0.112373038"
      ]
    ],
    [
      [
        "0.548766226",
        "0.588861468"
      ],
      [
        "0.973532366",
        "0.279430519"
      ],
      [
        "-0.636976794",
        "0.334998303"
      ],
      [
        "0.372408485",
        "0.592522483"
      ],
      [
        "0.999499953",
        "-0.344122967"
      ],
      [
        "0.755743316",
        "-0.181422596"
      ]
    ],
    [
      [
        "0.754090933",
        "0.468543339"
      ],
      [
        "-0.089391185",
        "-0.250902723"
      ],
      [
        "-0.449010902",
        "0.874992037"
      ],
      [
        "0.241640636",
        "-0.305550889"
      ],
      [
        "0.121046999",
        "0.878027413"
      ],
      [
        "0.328567244",
        "0.999499996"
      ]
    ]
  ]
}

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": 10,
  "d": 6
}

当前第一名的答案

{
  "vectors": [
    [
      [
        "0.999499563",
        "-0.199572763"
      ],
      [
        "0.447251077",
        "0.544147372"
      ],
      [
        "-0.201225674",
        "-0.200923148"
      ],
      [
        "0.827632643",
        "-0.009505971"
      ],
      [
        "0.183630022",
        "-0.590664826"
      ],
      [
        "-0.857751470",
        "0.840445603"
      ]
    ],
    [
      [
        "-0.631551936",
        "0.576236980"
      ],
      [
        "0.157957888",
        "-0.889323906"
      ],
      [
        "-0.064480125",
        "-0.457339132"
      ],
      [
        "0.312570108",
        "-0.219637271"
      ],
      [
        "0.999498778",
        "-0.434655746"
      ],
      [
        "-0.184185466",
        "-0.379671753"
      ]
    ],
    [
      [
        "-0.107739484",
        "-0.372929224"
      ],
      [
        "-0.682380717",
        "-0.634242056"
      ],
      [
        "-0.022807240",
        "0.299100304"
      ],
      [
        "0.522167562",
        "0.035576823"
      ],
      [
        "0.755496832",
        "0.489289283"
      ],
      [
        "-0.999500321",
        "0.408357269"
      ]
    ],
    [
      [
        "0.085609340",
        "0.334557121"
      ],
      [
        "0.212163855",
        "-0.032144926"
      ],
      [
        "-0.419079037",
        "0.999500149"
      ],
      [
        "-0.916167037",
        "0.638067645"
      ],
      [
        "-0.061269982",
        "0.041415189"
      ],
      [
        "-0.177171120",
        "-0.011015412"
      ]
    ],
    [
      [
        "0.475662476",
        "-0.129303141"
      ],
      [
        "-0.288035533",
        "0.578535209"
      ],
      [
        "0.150512997",
        "0.005309555"
      ],
      [
        "-0.029899101",
        "-0.781459188"
      ],
      [
        "0.302919024",
        "0.999499880"
      ],
      [
        "-0.415374760",
        "-0.532983748"
      ]
    ],
    [
      [
        "-0.465632395",
        "-0.321423840"
      ],
      [
        "-0.860847307",
        "0.868039111"
      ],
      [
        "-0.137602902",
        "0.020983683"
      ],
      [
        "0.486166011",
        "0.999500208"
      ],
      [
        "-0.096529674",
        "-0.234682355"
      ],
      [
        "0.260624277",
        "-0.075591580"
      ]
    ],
    [
      [
        "-0.399270392",
        "0.171133064"
      ],
      [
        "-0.521923524",
        "0.148353816"
      ],
      [
        "-0.485164438",
        "0.999500208"
      ],
      [
        "0.326058948",
        "-0.534183683"
      ],
      [
        "0.435506251",
        "-0.203446001"
      ],
      [
        "0.240046744",
        "-0.556421607"
      ]
    ],
    [
      [
        "-0.655431546",
        "-0.999500928"
      ],
      [
        "0.292935242",
        "0.246177527"
      ],
      [
        "-0.503815266",
        "0.713537922"
      ],
      [
        "0.086975686",
        "0.077165556"
      ],
      [
        "-0.219811237",
        "-0.164700604"
      ],
      [
        "-0.667966908",
        "-0.112373038"
      ]
    ],
    [
      [
        "0.548766226",
        "0.588861468"
      ],
      [
        "0.973532366",
        "0.279430519"
      ],
      [
        "-0.636976794",
        "0.334998303"
      ],
      [
        "0.372408485",
        "0.592522483"
      ],
      [
        "0.999499953",
        "-0.344122967"
      ],
      [
        "0.755743316",
        "-0.181422596"
      ]
    ],
    [
      [
        "0.754090933",
        "0.468543339"
      ],
      [
        "-0.089391185",
        "-0.250902723"
      ],
      [
        "-0.449010902",
        "0.874992037"
      ],
      [
        "0.241640636",
        "-0.305550889"
      ],
      [
        "0.121046999",
        "0.878027413"
      ],
      [
        "0.328567244",
        "0.999499996"
      ]
    ]
  ]
}

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。