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 = 4, n = 9

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 = 4, n = 9

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	[0.401850115, 0.401850125]	见来源署名	github.com ↗
本站纪录	0.401850121	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9 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（4 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
0.40185012049→0.40185012042
2026-10-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.401850120494→0.401850120492
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.401850120496→0.401850120494
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.401850122→0.401850121
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": [
    [
      [
        "0.272911897",
        "0.854746890"
      ],
      [
        "0.677574206",
        "-0.021190766"
      ],
      [
        "0.199849069",
        "0.160045053"
      ],
      [
        "0.821752371",
        "0.999499860"
      ]
    ],
    [
      [
        "-0.059838190",
        "0.743051642"
      ],
      [
        "-0.075909675",
        "-0.298217343"
      ],
      [
        "0.079743009",
        "-0.078971251"
      ],
      [
        "-0.999498657",
        "-0.541301266"
      ]
    ],
    [
      [
        "0.670265048",
        "0.427166631"
      ],
      [
        "0.999498643",
        "-0.637010890"
      ],
      [
        "0.833532178",
        "0.449710734"
      ],
      [
        "-0.298128124",
        "-0.485777262"
      ]
    ],
    [
      [
        "-0.306736402",
        "0.898068074"
      ],
      [
        "0.108607162",
        "-0.999499674"
      ],
      [
        "0.269220173",
        "-0.496953485"
      ],
      [
        "0.094374068",
        "0.190914380"
      ]
    ],
    [
      [
        "0.586335878",
        "-0.748948812"
      ],
      [
        "-0.166111888",
        "0.263051178"
      ],
      [
        "0.999499756",
        "0.033172814"
      ],
      [
        "0.233759675",
        "-0.057983304"
      ]
    ],
    [
      [
        "0.084589679",
        "0.336284223"
      ],
      [
        "-0.809781375",
        "0.350549489"
      ],
      [
        "0.598811255",
        "-0.999499929"
      ],
      [
        "-0.263388584",
        "-0.603384525"
      ]
    ],
    [
      [
        "-0.262424181",
        "0.035909886"
      ],
      [
        "-0.462267141",
        "0.295453287"
      ],
      [
        "0.223491544",
        "0.999499903"
      ],
      [
        "-0.177717346",
        "-0.589021776"
      ]
    ],
    [
      [
        "0.185814164",
        "0.794957205"
      ],
      [
        "-0.973031164",
        "0.999500362"
      ],
      [
        "0.403756641",
        "0.133374796"
      ],
      [
        "0.210526342",
        "0.518489834"
      ]
    ],
    [
      [
        "-0.402105899",
        "-0.716717987"
      ],
      [
        "-0.255029729",
        "-0.256250918"
      ],
      [
        "0.873239437",
        "0.466369335"
      ],
      [
        "-0.462423052",
        "0.999502001"
      ]
    ]
  ]
}

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": 9,
  "d": 4
}

当前第一名的答案

{
  "vectors": [
    [
      [
        "0.272911897",
        "0.854746890"
      ],
      [
        "0.677574206",
        "-0.021190766"
      ],
      [
        "0.199849069",
        "0.160045053"
      ],
      [
        "0.821752371",
        "0.999499860"
      ]
    ],
    [
      [
        "-0.059838190",
        "0.743051642"
      ],
      [
        "-0.075909675",
        "-0.298217343"
      ],
      [
        "0.079743009",
        "-0.078971251"
      ],
      [
        "-0.999498657",
        "-0.541301266"
      ]
    ],
    [
      [
        "0.670265048",
        "0.427166631"
      ],
      [
        "0.999498643",
        "-0.637010890"
      ],
      [
        "0.833532178",
        "0.449710734"
      ],
      [
        "-0.298128124",
        "-0.485777262"
      ]
    ],
    [
      [
        "-0.306736402",
        "0.898068074"
      ],
      [
        "0.108607162",
        "-0.999499674"
      ],
      [
        "0.269220173",
        "-0.496953485"
      ],
      [
        "0.094374068",
        "0.190914380"
      ]
    ],
    [
      [
        "0.586335878",
        "-0.748948812"
      ],
      [
        "-0.166111888",
        "0.263051178"
      ],
      [
        "0.999499756",
        "0.033172814"
      ],
      [
        "0.233759675",
        "-0.057983304"
      ]
    ],
    [
      [
        "0.084589679",
        "0.336284223"
      ],
      [
        "-0.809781375",
        "0.350549489"
      ],
      [
        "0.598811255",
        "-0.999499929"
      ],
      [
        "-0.263388584",
        "-0.603384525"
      ]
    ],
    [
      [
        "-0.262424181",
        "0.035909886"
      ],
      [
        "-0.462267141",
        "0.295453287"
      ],
      [
        "0.223491544",
        "0.999499903"
      ],
      [
        "-0.177717346",
        "-0.589021776"
      ]
    ],
    [
      [
        "0.185814164",
        "0.794957205"
      ],
      [
        "-0.973031164",
        "0.999500362"
      ],
      [
        "0.403756641",
        "0.133374796"
      ],
      [
        "0.210526342",
        "0.518489834"
      ]
    ],
    [
      [
        "-0.402105899",
        "-0.716717987"
      ],
      [
        "-0.255029729",
        "-0.256250918"
      ],
      [
        "0.873239437",
        "0.466369335"
      ],
      [
        "-0.462423052",
        "0.999502001"
      ]
    ]
  ]
}

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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