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 = 8

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 = 8

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	[0.295208075, 0.295208085]	见来源署名	github.com ↗
本站纪录	0.295208077	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8 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
0.2952080765→0.2952080764
2026-10-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.29520807653→0.29520807652
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.29520807655→0.29520807653
2026-09-01
StonebubuAI · Z.ai: GLM 5.3 Flash
0.295208078→0.295208077
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": [
    [
      [
        "0.520073693",
        "0.095331575"
      ],
      [
        "0.999999622",
        "-0.214529530"
      ],
      [
        "-0.396313189",
        "-0.783004109"
      ],
      [
        "0.732394977",
        "-0.222564354"
      ],
      [
        "0.206573379",
        "-0.215478139"
      ]
    ],
    [
      [
        "0.017964729",
        "-0.999498882"
      ],
      [
        "0.085568582",
        "0.385459261"
      ],
      [
        "0.162711568",
        "-0.473916161"
      ],
      [
        "0.881173036",
        "0.025619976"
      ],
      [
        "-0.084691218",
        "0.975360943"
      ]
    ],
    [
      [
        "-0.139686456",
        "-0.814723458"
      ],
      [
        "-0.729135341",
        "0.239419578"
      ],
      [
        "0.296074732",
        "0.258883667"
      ],
      [
        "0.453349228",
        "0.408456097"
      ],
      [
        "0.999999695",
        "-0.256974900"
      ]
    ],
    [
      [
        "-0.189184748",
        "0.183331227"
      ],
      [
        "-0.999499797",
        "0.051946483"
      ],
      [
        "0.207113108",
        "-0.929221990"
      ],
      [
        "-0.531073267",
        "0.143792192"
      ],
      [
        "0.168760948",
        "-0.592908934"
      ]
    ],
    [
      [
        "0.646919352",
        "0.076279876"
      ],
      [
        "0.123629209",
        "0.976555316"
      ],
      [
        "-0.436395494",
        "0.915044175"
      ],
      [
        "-0.999500091",
        "0.155286765"
      ],
      [
        "0.729121319",
        "0.080931430"
      ]
    ],
    [
      [
        "0.583365610",
        "-0.197198767"
      ],
      [
        "-0.869528548",
        "0.226762657"
      ],
      [
        "-0.495484025",
        "0.393106564"
      ],
      [
        "0.999999699",
        "-0.363047818"
      ],
      [
        "-0.280925335",
        "-0.330315204"
      ]
    ],
    [
      [
        "0.444702769",
        "-0.823981506"
      ],
      [
        "0.248490532",
        "0.198601473"
      ],
      [
        "-0.556269703",
        "-0.999999592"
      ],
      [
        "-0.752274833",
        "0.141696631"
      ],
      [
        "0.688929598",
        "0.145665771"
      ]
    ],
    [
      [
        "-0.978596002",
        "-0.058048427"
      ],
      [
        "-0.031269830",
        "-0.057332046"
      ],
      [
        "-0.484434949",
        "-0.119847714"
      ],
      [
        "-0.137114575",
        "0.657604767"
      ],
      [
        "-0.673064827",
        "0.999499967"
      ]
    ]
  ]
}

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": 8,
  "d": 5
}

当前第一名的答案

{
  "vectors": [
    [
      [
        "0.520073693",
        "0.095331575"
      ],
      [
        "0.999999622",
        "-0.214529530"
      ],
      [
        "-0.396313189",
        "-0.783004109"
      ],
      [
        "0.732394977",
        "-0.222564354"
      ],
      [
        "0.206573379",
        "-0.215478139"
      ]
    ],
    [
      [
        "0.017964729",
        "-0.999498882"
      ],
      [
        "0.085568582",
        "0.385459261"
      ],
      [
        "0.162711568",
        "-0.473916161"
      ],
      [
        "0.881173036",
        "0.025619976"
      ],
      [
        "-0.084691218",
        "0.975360943"
      ]
    ],
    [
      [
        "-0.139686456",
        "-0.814723458"
      ],
      [
        "-0.729135341",
        "0.239419578"
      ],
      [
        "0.296074732",
        "0.258883667"
      ],
      [
        "0.453349228",
        "0.408456097"
      ],
      [
        "0.999999695",
        "-0.256974900"
      ]
    ],
    [
      [
        "-0.189184748",
        "0.183331227"
      ],
      [
        "-0.999499797",
        "0.051946483"
      ],
      [
        "0.207113108",
        "-0.929221990"
      ],
      [
        "-0.531073267",
        "0.143792192"
      ],
      [
        "0.168760948",
        "-0.592908934"
      ]
    ],
    [
      [
        "0.646919352",
        "0.076279876"
      ],
      [
        "0.123629209",
        "0.976555316"
      ],
      [
        "-0.436395494",
        "0.915044175"
      ],
      [
        "-0.999500091",
        "0.155286765"
      ],
      [
        "0.729121319",
        "0.080931430"
      ]
    ],
    [
      [
        "0.583365610",
        "-0.197198767"
      ],
      [
        "-0.869528548",
        "0.226762657"
      ],
      [
        "-0.495484025",
        "0.393106564"
      ],
      [
        "0.999999699",
        "-0.363047818"
      ],
      [
        "-0.280925335",
        "-0.330315204"
      ]
    ],
    [
      [
        "0.444702769",
        "-0.823981506"
      ],
      [
        "0.248490532",
        "0.198601473"
      ],
      [
        "-0.556269703",
        "-0.999999592"
      ],
      [
        "-0.752274833",
        "0.141696631"
      ],
      [
        "0.688929598",
        "0.145665771"
      ]
    ],
    [
      [
        "-0.978596002",
        "-0.058048427"
      ],
      [
        "-0.031269830",
        "-0.057332046"
      ],
      [
        "-0.484434949",
        "-0.119847714"
      ],
      [
        "-0.137114575",
        "0.657604767"
      ],
      [
        "-0.673064827",
        "0.999499967"
      ]
    ]
  ]
}

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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