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 = 12

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 = 12

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	[0.427742315, 0.427742325]	见来源署名	github.com ↗
本站纪录	0.42774232	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12 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（5 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
0.427742321→0.42774232
2026-10-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.42774232008→0.42774232007
2026-09-07
StonebubuAI · Z.ai: GLM 5.3 Flash
0.42774232009→0.42774232008
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.4277423201→0.4277423200
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.4277423209→0.4277423201
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": [
    [
      [
        "-0.032837873",
        "0.784887557"
      ],
      [
        "0.582157750",
        "0.066080348"
      ],
      [
        "-0.320398208",
        "-0.999499847"
      ],
      [
        "-0.331258438",
        "0.659381664"
      ]
    ],
    [
      [
        "0.746801166",
        "0.478786614"
      ],
      [
        "0.472334893",
        "0.040560867"
      ],
      [
        "0.999500375",
        "-0.657939412"
      ],
      [
        "0.438981538",
        "0.533369702"
      ]
    ],
    [
      [
        "0.498557023",
        "0.836427832"
      ],
      [
        "-0.474647632",
        "-0.483013231"
      ],
      [
        "-0.214207151",
        "0.999499872"
      ],
      [
        "-0.334233395",
        "-0.592593156"
      ]
    ],
    [
      [
        "0.132333198",
        "0.543472384"
      ],
      [
        "0.999500662",
        "0.237114002"
      ],
      [
        "-0.059159601",
        "0.010947934"
      ],
      [
        "0.501227738",
        "-0.115830550"
      ]
    ],
    [
      [
        "0.562979727",
        "-0.999501670"
      ],
      [
        "0.119392330",
        "-0.932135746"
      ],
      [
        "0.866452393",
        "0.270544990"
      ],
      [
        "0.614777104",
        "0.693772513"
      ]
    ],
    [
      [
        "0.059863600",
        "0.933253129"
      ],
      [
        "0.262510032",
        "-0.999500029"
      ],
      [
        "0.322366005",
        "0.038052137"
      ],
      [
        "-0.786442583",
        "0.321322665"
      ]
    ],
    [
      [
        "0.999499635",
        "-0.201165798"
      ],
      [
        "0.384169520",
        "0.745392330"
      ],
      [
        "0.635860709",
        "0.629696306"
      ],
      [
        "0.810838205",
        "-0.253357489"
      ]
    ],
    [
      [
        "-0.813017982",
        "0.293079724"
      ],
      [
        "-0.462989650",
        "0.287022242"
      ],
      [
        "-0.112450146",
        "0.677587380"
      ],
      [
        "0.999500333",
        "-0.202785608"
      ]
    ],
    [
      [
        "0.875222992",
        "0.049611226"
      ],
      [
        "0.193958766",
        "0.999501538"
      ],
      [
        "-0.268541780",
        "-0.527730952"
      ],
      [
        "-0.085353065",
        "0.257714886"
      ]
    ],
    [
      [
        "-0.702215088",
        "-0.283013799"
      ],
      [
        "-0.055450613",
        "-0.481679665"
      ],
      [
        "0.785968780",
        "0.002079711"
      ],
      [
        "-0.537117243",
        "0.999499979"
      ]
    ],
    [
      [
        "-0.790650571",
        "0.116019669"
      ],
      [
        "0.803878836",
        "0.058792072"
      ],
      [
        "-0.568299963",
        "0.388338285"
      ],
      [
        "-0.330813539",
        "0.999499097"
      ]
    ],
    [
      [
        "0.183911658",
        "0.815032516"
      ],
      [
        "-0.179584253",
        "-0.621831630"
      ],
      [
        "-0.170893456",
        "-0.999499280"
      ],
      [
        "-0.158086262",
        "-0.707025025"
      ]
    ]
  ]
}

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": 12,
  "d": 4
}

当前第一名的答案

{
  "vectors": [
    [
      [
        "-0.032837873",
        "0.784887557"
      ],
      [
        "0.582157750",
        "0.066080348"
      ],
      [
        "-0.320398208",
        "-0.999499847"
      ],
      [
        "-0.331258438",
        "0.659381664"
      ]
    ],
    [
      [
        "0.746801166",
        "0.478786614"
      ],
      [
        "0.472334893",
        "0.040560867"
      ],
      [
        "0.999500375",
        "-0.657939412"
      ],
      [
        "0.438981538",
        "0.533369702"
      ]
    ],
    [
      [
        "0.498557023",
        "0.836427832"
      ],
      [
        "-0.474647632",
        "-0.483013231"
      ],
      [
        "-0.214207151",
        "0.999499872"
      ],
      [
        "-0.334233395",
        "-0.592593156"
      ]
    ],
    [
      [
        "0.132333198",
        "0.543472384"
      ],
      [
        "0.999500662",
        "0.237114002"
      ],
      [
        "-0.059159601",
        "0.010947934"
      ],
      [
        "0.501227738",
        "-0.115830550"
      ]
    ],
    [
      [
        "0.562979727",
        "-0.999501670"
      ],
      [
        "0.119392330",
        "-0.932135746"
      ],
      [
        "0.866452393",
        "0.270544990"
      ],
      [
        "0.614777104",
        "0.693772513"
      ]
    ],
    [
      [
        "0.059863600",
        "0.933253129"
      ],
      [
        "0.262510032",
        "-0.999500029"
      ],
      [
        "0.322366005",
        "0.038052137"
      ],
      [
        "-0.786442583",
        "0.321322665"
      ]
    ],
    [
      [
        "0.999499635",
        "-0.201165798"
      ],
      [
        "0.384169520",
        "0.745392330"
      ],
      [
        "0.635860709",
        "0.629696306"
      ],
      [
        "0.810838205",
        "-0.253357489"
      ]
    ],
    [
      [
        "-0.813017982",
        "0.293079724"
      ],
      [
        "-0.462989650",
        "0.287022242"
      ],
      [
        "-0.112450146",
        "0.677587380"
      ],
      [
        "0.999500333",
        "-0.202785608"
      ]
    ],
    [
      [
        "0.875222992",
        "0.049611226"
      ],
      [
        "0.193958766",
        "0.999501538"
      ],
      [
        "-0.268541780",
        "-0.527730952"
      ],
      [
        "-0.085353065",
        "0.257714886"
      ]
    ],
    [
      [
        "-0.702215088",
        "-0.283013799"
      ],
      [
        "-0.055450613",
        "-0.481679665"
      ],
      [
        "0.785968780",
        "0.002079711"
      ],
      [
        "-0.537117243",
        "0.999499979"
      ]
    ],
    [
      [
        "-0.790650571",
        "0.116019669"
      ],
      [
        "0.803878836",
        "0.058792072"
      ],
      [
        "-0.568299963",
        "0.388338285"
      ],
      [
        "-0.330813539",
        "0.999499097"
      ]
    ],
    [
      [
        "0.183911658",
        "0.815032516"
      ],
      [
        "-0.179584253",
        "-0.621831630"
      ],
      [
        "-0.170893456",
        "-0.999499280"
      ],
      [
        "-0.158086262",
        "-0.707025025"
      ]
    ]
  ]
}

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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