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 = 6, n = 13

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 = 6, n = 13

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	[0.311824555, 0.311824565]	见来源署名	github.com ↗
本站纪录	0.311824564	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13 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
0.3118245631→0.3118245630
2026-10-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.31182456318→0.31182456317
2026-09-03
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.3118245632→0.3118245631
2026-09-01
StonebubuAI · Z.ai: GLM 5.3 Flash
0.311824565→0.311824564
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": [
    [
      [
        "0.078495999",
        "-0.999501411"
      ],
      [
        "0.187054141",
        "0.390263905"
      ],
      [
        "0.339264117",
        "-0.738305877"
      ],
      [
        "0.401707085",
        "0.624812104"
      ],
      [
        "-0.606433371",
        "-0.560582351"
      ],
      [
        "-0.218335645",
        "0.149132984"
      ]
    ],
    [
      [
        "-0.999993765",
        "-0.242222377"
      ],
      [
        "-0.384613684",
        "-0.811531948"
      ],
      [
        "-0.460612866",
        "0.470555868"
      ],
      [
        "0.455581894",
        "0.781388008"
      ],
      [
        "0.143797539",
        "0.174210689"
      ],
      [
        "-0.314465751",
        "-0.591275320"
      ]
    ],
    [
      [
        "-0.999994050",
        "0.484074679"
      ],
      [
        "0.358759659",
        "0.468043877"
      ],
      [
        "0.436065860",
        "-0.224372188"
      ],
      [
        "0.940469748",
        "-0.134436900"
      ],
      [
        "-0.090689767",
        "-0.887767686"
      ],
      [
        "0.552273186",
        "-0.473166257"
      ]
    ],
    [
      [
        "-0.308922459",
        "-0.118063474"
      ],
      [
        "0.999994252",
        "-0.872320850"
      ],
      [
        "-0.232273499",
        "0.958257785"
      ],
      [
        "-0.136664323",
        "-0.072882418"
      ],
      [
        "-0.120638702",
        "0.214686486"
      ],
      [
        "-0.901673690",
        "0.451048799"
      ]
    ],
    [
      [
        "-0.295734757",
        "0.105762318"
      ],
      [
        "0.398282094",
        "-0.076065642"
      ],
      [
        "0.120074965",
        "-0.273507452"
      ],
      [
        "-0.192595505",
        "-0.426339327"
      ],
      [
        "0.999995232",
        "0.027839567"
      ],
      [
        "0.745880215",
        "0.699763677"
      ]
    ],
    [
      [
        "-0.783698060",
        "-0.437244830"
      ],
      [
        "-0.549665723",
        "0.814027142"
      ],
      [
        "-0.954687173",
        "0.940468607"
      ],
      [
        "-0.996127762",
        "-0.060231744"
      ],
      [
        "0.182810764",
        "0.104358808"
      ],
      [
        "0.999996959",
        "-0.782755854"
      ]
    ],
    [
      [
        "0.369056865",
        "-0.999500958"
      ],
      [
        "-0.158594229",
        "-0.362718644"
      ],
      [
        "-0.137620809",
        "0.590273066"
      ],
      [
        "-0.388511096",
        "-0.601083126"
      ],
      [
        "0.353307322",
        "-0.207488495"
      ],
      [
        "-0.411358381",
        "-0.979255840"
      ]
    ],
    [
      [
        "0.999996476",
        "-0.352493232"
      ],
      [
        "0.119517752",
        "-0.306441340"
      ],
      [
        "0.677769192",
        "0.593955642"
      ],
      [
        "0.241702455",
        "-0.113153047"
      ],
      [
        "-0.618710760",
        "0.228940884"
      ],
      [
        "0.231103578",
        "0.161311742"
      ]
    ],
    [
      [
        "0.063356650",
        "-0.356029206"
      ],
      [
        "-0.342990275",
        "0.659575757"
      ],
      [
        "-0.218435724",
        "0.558956446"
      ],
      [
        "0.999500498",
        "-0.575230097"
      ],
      [
        "-0.505248194",
        "0.042954909"
      ],
      [
        "0.421466092",
        "-0.098525604"
      ]
    ],
    [
      [
        "0.070298486",
        "0.572116766"
      ],
      [
        "-0.559026329",
        "-0.569393439"
      ],
      [
        "-0.512346272",
        "0.547619159"
      ],
      [
        "-0.508224069",
        "0.862553832"
      ],
      [
        "-0.999993280",
        "0.756442488"
      ],
      [
        "0.837506625",
        "0.095643200"
      ]
    ],
    [
      [
        "0.102696500",
        "0.564341472"
      ],
      [
        "-0.169794356",
        "0.999500922"
      ],
      [
        "-0.108036438",
        "-0.201361934"
      ],
      [
        "-0.718558271",
        "0.286169248"
      ],
      [
        "0.311698374",
        "0.828218148"
      ],
      [
        "-0.183885530",
        "0.108426495"
      ]
    ],
    [
      [
        "0.222037334",
        "0.164297823"
      ],
      [
        "0.131347552",
        "-0.196474022"
      ],
      [
        "-0.527897343",
        "-0.999995408"
      ],
      [
        "0.458269356",
        "0.547286764"
      ],
      [
        "0.144197132",
        "0.979786060"
      ],
      [
        "-0.523220840",
        "-0.478969750"
      ]
    ],
    [
      [
        "-0.305978999",
        "-0.353434452"
      ],
      [
        "0.999996443",
        "0.557839713"
      ],
      [
        "0.019044388",
        "0.793998208"
      ],
      [
        "0.014186021",
        "0.492307534"
      ],
      [
        "-0.536884837",
        "0.509239319"
      ],
      [
        "0.741818938",
        "0.592120795"
      ]
    ]
  ]
}

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": 13,
  "d": 6
}

当前第一名的答案

{
  "vectors": [
    [
      [
        "0.078495999",
        "-0.999501411"
      ],
      [
        "0.187054141",
        "0.390263905"
      ],
      [
        "0.339264117",
        "-0.738305877"
      ],
      [
        "0.401707085",
        "0.624812104"
      ],
      [
        "-0.606433371",
        "-0.560582351"
      ],
      [
        "-0.218335645",
        "0.149132984"
      ]
    ],
    [
      [
        "-0.999993765",
        "-0.242222377"
      ],
      [
        "-0.384613684",
        "-0.811531948"
      ],
      [
        "-0.460612866",
        "0.470555868"
      ],
      [
        "0.455581894",
        "0.781388008"
      ],
      [
        "0.143797539",
        "0.174210689"
      ],
      [
        "-0.314465751",
        "-0.591275320"
      ]
    ],
    [
      [
        "-0.999994050",
        "0.484074679"
      ],
      [
        "0.358759659",
        "0.468043877"
      ],
      [
        "0.436065860",
        "-0.224372188"
      ],
      [
        "0.940469748",
        "-0.134436900"
      ],
      [
        "-0.090689767",
        "-0.887767686"
      ],
      [
        "0.552273186",
        "-0.473166257"
      ]
    ],
    [
      [
        "-0.308922459",
        "-0.118063474"
      ],
      [
        "0.999994252",
        "-0.872320850"
      ],
      [
        "-0.232273499",
        "0.958257785"
      ],
      [
        "-0.136664323",
        "-0.072882418"
      ],
      [
        "-0.120638702",
        "0.214686486"
      ],
      [
        "-0.901673690",
        "0.451048799"
      ]
    ],
    [
      [
        "-0.295734757",
        "0.105762318"
      ],
      [
        "0.398282094",
        "-0.076065642"
      ],
      [
        "0.120074965",
        "-0.273507452"
      ],
      [
        "-0.192595505",
        "-0.426339327"
      ],
      [
        "0.999995232",
        "0.027839567"
      ],
      [
        "0.745880215",
        "0.699763677"
      ]
    ],
    [
      [
        "-0.783698060",
        "-0.437244830"
      ],
      [
        "-0.549665723",
        "0.814027142"
      ],
      [
        "-0.954687173",
        "0.940468607"
      ],
      [
        "-0.996127762",
        "-0.060231744"
      ],
      [
        "0.182810764",
        "0.104358808"
      ],
      [
        "0.999996959",
        "-0.782755854"
      ]
    ],
    [
      [
        "0.369056865",
        "-0.999500958"
      ],
      [
        "-0.158594229",
        "-0.362718644"
      ],
      [
        "-0.137620809",
        "0.590273066"
      ],
      [
        "-0.388511096",
        "-0.601083126"
      ],
      [
        "0.353307322",
        "-0.207488495"
      ],
      [
        "-0.411358381",
        "-0.979255840"
      ]
    ],
    [
      [
        "0.999996476",
        "-0.352493232"
      ],
      [
        "0.119517752",
        "-0.306441340"
      ],
      [
        "0.677769192",
        "0.593955642"
      ],
      [
        "0.241702455",
        "-0.113153047"
      ],
      [
        "-0.618710760",
        "0.228940884"
      ],
      [
        "0.231103578",
        "0.161311742"
      ]
    ],
    [
      [
        "0.063356650",
        "-0.356029206"
      ],
      [
        "-0.342990275",
        "0.659575757"
      ],
      [
        "-0.218435724",
        "0.558956446"
      ],
      [
        "0.999500498",
        "-0.575230097"
      ],
      [
        "-0.505248194",
        "0.042954909"
      ],
      [
        "0.421466092",
        "-0.098525604"
      ]
    ],
    [
      [
        "0.070298486",
        "0.572116766"
      ],
      [
        "-0.559026329",
        "-0.569393439"
      ],
      [
        "-0.512346272",
        "0.547619159"
      ],
      [
        "-0.508224069",
        "0.862553832"
      ],
      [
        "-0.999993280",
        "0.756442488"
      ],
      [
        "0.837506625",
        "0.095643200"
      ]
    ],
    [
      [
        "0.102696500",
        "0.564341472"
      ],
      [
        "-0.169794356",
        "0.999500922"
      ],
      [
        "-0.108036438",
        "-0.201361934"
      ],
      [
        "-0.718558271",
        "0.286169248"
      ],
      [
        "0.311698374",
        "0.828218148"
      ],
      [
        "-0.183885530",
        "0.108426495"
      ]
    ],
    [
      [
        "0.222037334",
        "0.164297823"
      ],
      [
        "0.131347552",
        "-0.196474022"
      ],
      [
        "-0.527897343",
        "-0.999995408"
      ],
      [
        "0.458269356",
        "0.547286764"
      ],
      [
        "0.144197132",
        "0.979786060"
      ],
      [
        "-0.523220840",
        "-0.478969750"
      ]
    ],
    [
      [
        "-0.305978999",
        "-0.353434452"
      ],
      [
        "0.999996443",
        "0.557839713"
      ],
      [
        "0.019044388",
        "0.793998208"
      ],
      [
        "0.014186021",
        "0.492307534"
      ],
      [
        "-0.536884837",
        "0.509239319"
      ],
      [
        "0.741818938",
        "0.592120795"
      ]
    ]
  ]
}

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

DISCUSSION

讨论区

欢迎交流解题思路、分享构造方法或探讨卡壳之处。所有登录用户均可发帖；发言使用与排行榜一致的公开署名，并附带唯一注册编号（#UID）以防冒名。新发言经审核后公开。

暂无讨论，欢迎发表第一篇帖子。