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 = 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 = 5, 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	[0.320117135, 0.320117145]	见来源署名	github.com ↗
本站纪录	0.320117145	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
0.3201171446→0.3201171445
2026-10-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.320117144644→0.320117144641
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.32011714466→0.32011714464
2026-09-01
StonebubuAI · Z.ai: GLM 5.3 Flash
0.320117146→0.320117145
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": [
    [
      [
        "0.212445571",
        "-0.424357357"
      ],
      [
        "-0.083376378",
        "-0.927177744"
      ],
      [
        "-0.156351429",
        "0.134218508"
      ],
      [
        "-0.287789494",
        "-0.999500045"
      ],
      [
        "-0.198169172",
        "-0.569409535"
      ]
    ],
    [
      [
        "0.085987912",
        "0.359824704"
      ],
      [
        "0.149761995",
        "-0.009811995"
      ],
      [
        "-0.999499810",
        "0.183150703"
      ],
      [
        "0.046116363",
        "0.312309741"
      ],
      [
        "0.426535822",
        "-0.340518416"
      ]
    ],
    [
      [
        "0.032736923",
        "-0.733628679"
      ],
      [
        "0.332376859",
        "-0.299319436"
      ],
      [
        "0.968534387",
        "0.339793725"
      ],
      [
        "-0.241265957",
        "-0.132002979"
      ],
      [
        "0.999500228",
        "-0.547778629"
      ]
    ],
    [
      [
        "-0.717933902",
        "-0.460039618"
      ],
      [
        "-0.353084562",
        "0.778400676"
      ],
      [
        "0.660247109",
        "0.090704554"
      ],
      [
        "0.999500046",
        "-0.126945123"
      ],
      [
        "0.348752023",
        "-0.266561302"
      ]
    ],
    [
      [
        "0.530796784",
        "0.106794541"
      ],
      [
        "-0.128441011",
        "-0.999995074"
      ],
      [
        "0.192525279",
        "0.054448266"
      ],
      [
        "0.871877844",
        "-0.117107792"
      ],
      [
        "-0.302564874",
        "0.813243688"
      ]
    ],
    [
      [
        "-0.508644253",
        "-0.707763208"
      ],
      [
        "0.790785879",
        "0.227047913"
      ],
      [
        "-0.295519694",
        "-0.339774314"
      ],
      [
        "0.876328011",
        "-0.425004128"
      ],
      [
        "-0.530058955",
        "0.999995213"
      ]
    ],
    [
      [
        "-0.308733186",
        "-0.652599116"
      ],
      [
        "0.999499512",
        "-0.297039219"
      ],
      [
        "0.268877875",
        "0.510712424"
      ],
      [
        "-0.888623760",
        "-0.077608915"
      ],
      [
        "-0.434382960",
        "0.345543456"
      ]
    ],
    [
      [
        "0.268286488",
        "-0.856150635"
      ],
      [
        "-0.470826388",
        "-0.118753141"
      ],
      [
        "-0.999995194",
        "-0.533929176"
      ],
      [
        "0.092635376",
        "-0.040263747"
      ],
      [
        "-0.195664710",
        "-0.067834578"
      ]
    ],
    [
      [
        "-0.999994950",
        "-0.539361506"
      ],
      [
        "0.396867047",
        "-0.827914458"
      ],
      [
        "-0.478436170",
        "0.151422024"
      ],
      [
        "0.979183687",
        "0.491094734"
      ],
      [
        "0.130515393",
        "-0.513704725"
      ]
    ]
  ]
}

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": 5
}

当前第一名的答案

{
  "vectors": [
    [
      [
        "0.212445571",
        "-0.424357357"
      ],
      [
        "-0.083376378",
        "-0.927177744"
      ],
      [
        "-0.156351429",
        "0.134218508"
      ],
      [
        "-0.287789494",
        "-0.999500045"
      ],
      [
        "-0.198169172",
        "-0.569409535"
      ]
    ],
    [
      [
        "0.085987912",
        "0.359824704"
      ],
      [
        "0.149761995",
        "-0.009811995"
      ],
      [
        "-0.999499810",
        "0.183150703"
      ],
      [
        "0.046116363",
        "0.312309741"
      ],
      [
        "0.426535822",
        "-0.340518416"
      ]
    ],
    [
      [
        "0.032736923",
        "-0.733628679"
      ],
      [
        "0.332376859",
        "-0.299319436"
      ],
      [
        "0.968534387",
        "0.339793725"
      ],
      [
        "-0.241265957",
        "-0.132002979"
      ],
      [
        "0.999500228",
        "-0.547778629"
      ]
    ],
    [
      [
        "-0.717933902",
        "-0.460039618"
      ],
      [
        "-0.353084562",
        "0.778400676"
      ],
      [
        "0.660247109",
        "0.090704554"
      ],
      [
        "0.999500046",
        "-0.126945123"
      ],
      [
        "0.348752023",
        "-0.266561302"
      ]
    ],
    [
      [
        "0.530796784",
        "0.106794541"
      ],
      [
        "-0.128441011",
        "-0.999995074"
      ],
      [
        "0.192525279",
        "0.054448266"
      ],
      [
        "0.871877844",
        "-0.117107792"
      ],
      [
        "-0.302564874",
        "0.813243688"
      ]
    ],
    [
      [
        "-0.508644253",
        "-0.707763208"
      ],
      [
        "0.790785879",
        "0.227047913"
      ],
      [
        "-0.295519694",
        "-0.339774314"
      ],
      [
        "0.876328011",
        "-0.425004128"
      ],
      [
        "-0.530058955",
        "0.999995213"
      ]
    ],
    [
      [
        "-0.308733186",
        "-0.652599116"
      ],
      [
        "0.999499512",
        "-0.297039219"
      ],
      [
        "0.268877875",
        "0.510712424"
      ],
      [
        "-0.888623760",
        "-0.077608915"
      ],
      [
        "-0.434382960",
        "0.345543456"
      ]
    ],
    [
      [
        "0.268286488",
        "-0.856150635"
      ],
      [
        "-0.470826388",
        "-0.118753141"
      ],
      [
        "-0.999995194",
        "-0.533929176"
      ],
      [
        "0.092635376",
        "-0.040263747"
      ],
      [
        "-0.195664710",
        "-0.067834578"
      ]
    ],
    [
      [
        "-0.999994950",
        "-0.539361506"
      ],
      [
        "0.396867047",
        "-0.827914458"
      ],
      [
        "-0.478436170",
        "0.151422024"
      ],
      [
        "0.979183687",
        "0.491094734"
      ],
      [
        "0.130515393",
        "-0.513704725"
      ]
    ]
  ]
}

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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