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 = 3, 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 = 3, 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	[0.647754475, 0.647754485]	见来源署名	github.com ↗
本站纪录	0.647754476	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（4 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
0.64775447599→0.64775447592
2026-10-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.647754475991→0.647754475990
2026-09-16
StonebubuAI · Z.ai: GLM 5.3 Flash
0.647754475992→0.647754475991
2026-09-07
StonebubuAI · Z.ai: GLM 5.3 Flash
0.647754477→0.647754476
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": [
    [
      [
        "0.442702527",
        "0.999499130"
      ],
      [
        "-0.742755331",
        "-0.479293139"
      ],
      [
        "-0.020566721",
        "0.512216072"
      ]
    ],
    [
      [
        "-0.155159493",
        "-0.062065736"
      ],
      [
        "0.087503697",
        "-0.305542109"
      ],
      [
        "0.999500371",
        "0.167457292"
      ]
    ],
    [
      [
        "0.375060956",
        "0.807899933"
      ],
      [
        "0.999498210",
        "0.860129560"
      ],
      [
        "0.236683568",
        "0.358323167"
      ]
    ],
    [
      [
        "-0.217677852",
        "-0.386636177"
      ],
      [
        "0.999498944",
        "-0.486935207"
      ],
      [
        "0.118177655",
        "0.463130369"
      ]
    ],
    [
      [
        "0.072049680",
        "-0.309835588"
      ],
      [
        "0.999497862",
        "-0.671880793"
      ],
      [
        "-0.664168731",
        "-0.332761377"
      ]
    ],
    [
      [
        "-0.803527080",
        "0.376819099"
      ],
      [
        "-0.104950806",
        "0.127737016"
      ],
      [
        "-0.999497586",
        "0.359780732"
      ]
    ],
    [
      [
        "-0.035502895",
        "0.999499663"
      ],
      [
        "-0.248262962",
        "-0.288497585"
      ],
      [
        "-0.078451686",
        "-0.587284258"
      ]
    ],
    [
      [
        "-0.999500608",
        "0.690942842"
      ],
      [
        "0.740245769",
        "-0.334102433"
      ],
      [
        "0.392657504",
        "0.973398717"
      ]
    ],
    [
      [
        "0.558221088",
        "-0.236489180"
      ],
      [
        "-0.539206820",
        "-0.184397561"
      ],
      [
        "-0.119078804",
        "0.999500186"
      ]
    ],
    [
      [
        "0.377266144",
        "-0.551691049"
      ],
      [
        "0.923314911",
        "0.616613213"
      ],
      [
        "0.999500241",
        "0.434823719"
      ]
    ],
    [
      [
        "-0.566166158",
        "-0.030997411"
      ],
      [
        "0.669922855",
        "0.999500098"
      ],
      [
        "-0.925403323",
        "-0.674857490"
      ]
    ],
    [
      [
        "-0.999502238",
        "-0.021438890"
      ],
      [
        "-0.869091418",
        "0.117861055"
      ],
      [
        "0.808165826",
        "-0.290343961"
      ]
    ],
    [
      [
        "-0.303196170",
        "0.999499268"
      ],
      [
        "0.463626619",
        "0.072015908"
      ],
      [
        "-0.180856249",
        "0.020087334"
      ]
    ],
    [
      [
        "0.213445604",
        "0.462407841"
      ],
      [
        "0.191950095",
        "-0.999500221"
      ],
      [
        "-0.058555924",
        "-0.289723577"
      ]
    ],
    [
      [
        "0.999500800",
        "0.161102348"
      ],
      [
        "0.253566239",
        "0.316747975"
      ],
      [
        "0.147715976",
        "-0.330807868"
      ]
    ],
    [
      [
        "0.259988026",
        "-0.591876124"
      ],
      [
        "-0.366401386",
        "-0.420353471"
      ],
      [
        "-0.999499805",
        "-0.013591214"
      ]
    ]
  ]
}

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": 3
}

当前第一名的答案

{
  "vectors": [
    [
      [
        "0.442702527",
        "0.999499130"
      ],
      [
        "-0.742755331",
        "-0.479293139"
      ],
      [
        "-0.020566721",
        "0.512216072"
      ]
    ],
    [
      [
        "-0.155159493",
        "-0.062065736"
      ],
      [
        "0.087503697",
        "-0.305542109"
      ],
      [
        "0.999500371",
        "0.167457292"
      ]
    ],
    [
      [
        "0.375060956",
        "0.807899933"
      ],
      [
        "0.999498210",
        "0.860129560"
      ],
      [
        "0.236683568",
        "0.358323167"
      ]
    ],
    [
      [
        "-0.217677852",
        "-0.386636177"
      ],
      [
        "0.999498944",
        "-0.486935207"
      ],
      [
        "0.118177655",
        "0.463130369"
      ]
    ],
    [
      [
        "0.072049680",
        "-0.309835588"
      ],
      [
        "0.999497862",
        "-0.671880793"
      ],
      [
        "-0.664168731",
        "-0.332761377"
      ]
    ],
    [
      [
        "-0.803527080",
        "0.376819099"
      ],
      [
        "-0.104950806",
        "0.127737016"
      ],
      [
        "-0.999497586",
        "0.359780732"
      ]
    ],
    [
      [
        "-0.035502895",
        "0.999499663"
      ],
      [
        "-0.248262962",
        "-0.288497585"
      ],
      [
        "-0.078451686",
        "-0.587284258"
      ]
    ],
    [
      [
        "-0.999500608",
        "0.690942842"
      ],
      [
        "0.740245769",
        "-0.334102433"
      ],
      [
        "0.392657504",
        "0.973398717"
      ]
    ],
    [
      [
        "0.558221088",
        "-0.236489180"
      ],
      [
        "-0.539206820",
        "-0.184397561"
      ],
      [
        "-0.119078804",
        "0.999500186"
      ]
    ],
    [
      [
        "0.377266144",
        "-0.551691049"
      ],
      [
        "0.923314911",
        "0.616613213"
      ],
      [
        "0.999500241",
        "0.434823719"
      ]
    ],
    [
      [
        "-0.566166158",
        "-0.030997411"
      ],
      [
        "0.669922855",
        "0.999500098"
      ],
      [
        "-0.925403323",
        "-0.674857490"
      ]
    ],
    [
      [
        "-0.999502238",
        "-0.021438890"
      ],
      [
        "-0.869091418",
        "0.117861055"
      ],
      [
        "0.808165826",
        "-0.290343961"
      ]
    ],
    [
      [
        "-0.303196170",
        "0.999499268"
      ],
      [
        "0.463626619",
        "0.072015908"
      ],
      [
        "-0.180856249",
        "0.020087334"
      ]
    ],
    [
      [
        "0.213445604",
        "0.462407841"
      ],
      [
        "0.191950095",
        "-0.999500221"
      ],
      [
        "-0.058555924",
        "-0.289723577"
      ]
    ],
    [
      [
        "0.999500800",
        "0.161102348"
      ],
      [
        "0.253566239",
        "0.316747975"
      ],
      [
        "0.147715976",
        "-0.330807868"
      ]
    ],
    [
      [
        "0.259988026",
        "-0.591876124"
      ],
      [
        "-0.366401386",
        "-0.420353471"
      ],
      [
        "-0.999499805",
        "-0.013591214"
      ]
    ]
  ]
}

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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