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 = 14

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 = 14

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	[0.320316635, 0.320316645]	见来源署名	github.com ↗
本站纪录	0.320316638	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14 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（7 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
0.3203166379→0.3203166378
2026-10-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.320316637945→0.320316637943
2026-09-25
StonebubuAI · Z.ai: GLM 5.3 Flash
0.320316637947→0.320316637945
2026-09-05
StonebubuAI · Z.ai: GLM 5.3 Flash
0.32031663795→0.32031663794
2026-09-03
StonebubuAI · Z.ai: GLM 5.3 Flash
0.320316637959→0.320316637951
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.32031663796→0.32031663795
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.320316639→0.320316638
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": [
    [
      [
        "0.055079297",
        "-0.794308160"
      ],
      [
        "-0.764080635",
        "-0.006298055"
      ],
      [
        "0.073952790",
        "-0.413793859"
      ],
      [
        "-0.364423754",
        "0.848985422"
      ],
      [
        "-0.641875896",
        "0.999497412"
      ],
      [
        "-0.723029034",
        "-0.365759494"
      ]
    ],
    [
      [
        "0.001534602",
        "-0.320471608"
      ],
      [
        "0.583140932",
        "-0.118227114"
      ],
      [
        "-0.073425844",
        "-0.999498683"
      ],
      [
        "0.680627400",
        "-0.255866165"
      ],
      [
        "0.225715884",
        "-0.567845348"
      ],
      [
        "-0.352048028",
        "0.271667200"
      ]
    ],
    [
      [
        "-0.136563599",
        "0.457399975"
      ],
      [
        "-0.390783540",
        "-0.845792697"
      ],
      [
        "0.083522756",
        "-0.853711043"
      ],
      [
        "0.877596751",
        "-0.818612783"
      ],
      [
        "0.999500108",
        "0.339942002"
      ],
      [
        "0.563824028",
        "-0.356648067"
      ]
    ],
    [
      [
        "-0.167254035",
        "-0.103367053"
      ],
      [
        "0.601237046",
        "-0.408856883"
      ],
      [
        "-0.999503138",
        "0.497787153"
      ],
      [
        "-0.444689202",
        "0.188506667"
      ],
      [
        "0.307996299",
        "0.311297050"
      ],
      [
        "-0.970029816",
        "0.693517559"
      ]
    ],
    [
      [
        "0.244354174",
        "0.775316341"
      ],
      [
        "-0.115269752",
        "0.999499359"
      ],
      [
        "0.029023985",
        "0.107024228"
      ],
      [
        "0.109096209",
        "-0.130897337"
      ],
      [
        "-0.095473031",
        "0.242887997"
      ],
      [
        "-0.876872700",
        "0.169643392"
      ]
    ],
    [
      [
        "0.695788106",
        "-0.999500833"
      ],
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        "-0.188391732",
        "0.188555286"
      ],
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        "-0.894527976",
        "0.201763266"
      ],
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        "0.304111783",
        "0.027649968"
      ],
      [
        "-0.416030081",
        "0.238533066"
      ],
      [
        "-0.057291621",
        "0.577027370"
      ]
    ],
    [
      [
        "-0.999501648",
        "0.522294275"
      ],
      [
        "0.580590737",
        "0.037113526"
      ],
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        "-0.011759596",
        "0.285441888"
      ],
      [
        "0.241253020",
        "-0.249722984"
      ],
      [
        "-0.590047685",
        "-0.257186893"
      ],
      [
        "-0.594563955",
        "0.355298149"
      ]
    ],
    [
      [
        "0.766386830",
        "0.900253369"
      ],
      [
        "0.999498548",
        "0.274741780"
      ],
      [
        "-0.548773466",
        "0.189057389"
      ],
      [
        "-0.797898270",
        "0.531547782"
      ],
      [
        "-0.259680717",
        "0.289541380"
      ],
      [
        "0.274150021",
        "-0.616804503"
      ]
    ],
    [
      [
        "-0.051976817",
        "0.056555396"
      ],
      [
        "0.402563489",
        "-0.999498549"
      ],
      [
        "-0.025296930",
        "0.246523273"
      ],
      [
        "-0.574578960",
        "-0.619418960"
      ],
      [
        "0.288145543",
        "0.656582772"
      ],
      [
        "-0.086415454",
        "-0.550480844"
      ]
    ],
    [
      [
        "0.221631775",
        "-0.186776651"
      ],
      [
        "-0.429915508",
        "0.637733016"
      ],
      [
        "0.486456651",
        "0.457596238"
      ],
      [
        "-0.608200474",
        "-0.583014550"
      ],
      [
        "0.999499675",
        "-0.058717775"
      ],
      [
        "0.224188132",
        "0.545339928"
      ]
    ],
    [
      [
        "-0.635477198",
        "0.759666005"
      ],
      [
        "-0.172769285",
        "-0.093099868"
      ],
      [
        "-0.247207931",
        "-0.061222943"
      ],
      [
        "-0.182012953",
        "0.999503375"
      ],
      [
        "-0.062247544",
        "0.208981333"
      ],
      [
        "0.159955704",
        "0.211040819"
      ]
    ],
    [
      [
        "-0.212730803",
        "0.169055791"
      ],
      [
        "-0.088014023",
        "-0.190450117"
      ],
      [
        "0.852554276",
        "-0.908136129"
      ],
      [
        "-0.183129479",
        "-0.690486841"
      ],
      [
        "0.124688320",
        "0.831828316"
      ],
      [
        "-0.999499220",
        "0.333840879"
      ]
    ],
    [
      [
        "0.415734154",
        "0.236908204"
      ],
      [
        "-0.812714585",
        "-0.051661127"
      ],
      [
        "0.167856299",
        "0.677738078"
      ],
      [
        "0.452563783",
        "-0.561052799"
      ],
      [
        "-0.999499640",
        "-0.689632974"
      ],
      [
        "-0.393472045",
        "0.529641861"
      ]
    ],
    [
      [
        "0.056507894",
        "0.146961019"
      ],
      [
        "0.535719941",
        "0.437487389"
      ],
      [
        "-0.093995653",
        "0.999500053"
      ],
      [
        "-0.107288592",
        "-0.362163762"
      ],
      [
        "0.820377238",
        "-0.099973500"
      ],
      [
        "-0.597515711",
        "-0.773380866"
      ]
    ]
  ]
}

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": 14,
  "d": 6
}

当前第一名的答案

{
  "vectors": [
    [
      [
        "0.055079297",
        "-0.794308160"
      ],
      [
        "-0.764080635",
        "-0.006298055"
      ],
      [
        "0.073952790",
        "-0.413793859"
      ],
      [
        "-0.364423754",
        "0.848985422"
      ],
      [
        "-0.641875896",
        "0.999497412"
      ],
      [
        "-0.723029034",
        "-0.365759494"
      ]
    ],
    [
      [
        "0.001534602",
        "-0.320471608"
      ],
      [
        "0.583140932",
        "-0.118227114"
      ],
      [
        "-0.073425844",
        "-0.999498683"
      ],
      [
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      ],
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        "0.225715884",
        "-0.567845348"
      ],
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        "-0.352048028",
        "0.271667200"
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        "-0.853711043"
      ],
      [
        "0.877596751",
        "-0.818612783"
      ],
      [
        "0.999500108",
        "0.339942002"
      ],
      [
        "0.563824028",
        "-0.356648067"
      ]
    ],
    [
      [
        "-0.167254035",
        "-0.103367053"
      ],
      [
        "0.601237046",
        "-0.408856883"
      ],
      [
        "-0.999503138",
        "0.497787153"
      ],
      [
        "-0.444689202",
        "0.188506667"
      ],
      [
        "0.307996299",
        "0.311297050"
      ],
      [
        "-0.970029816",
        "0.693517559"
      ]
    ],
    [
      [
        "0.244354174",
        "0.775316341"
      ],
      [
        "-0.115269752",
        "0.999499359"
      ],
      [
        "0.029023985",
        "0.107024228"
      ],
      [
        "0.109096209",
        "-0.130897337"
      ],
      [
        "-0.095473031",
        "0.242887997"
      ],
      [
        "-0.876872700",
        "0.169643392"
      ]
    ],
    [
      [
        "0.695788106",
        "-0.999500833"
      ],
      [
        "-0.188391732",
        "0.188555286"
      ],
      [
        "-0.894527976",
        "0.201763266"
      ],
      [
        "0.304111783",
        "0.027649968"
      ],
      [
        "-0.416030081",
        "0.238533066"
      ],
      [
        "-0.057291621",
        "0.577027370"
      ]
    ],
    [
      [
        "-0.999501648",
        "0.522294275"
      ],
      [
        "0.580590737",
        "0.037113526"
      ],
      [
        "-0.011759596",
        "0.285441888"
      ],
      [
        "0.241253020",
        "-0.249722984"
      ],
      [
        "-0.590047685",
        "-0.257186893"
      ],
      [
        "-0.594563955",
        "0.355298149"
      ]
    ],
    [
      [
        "0.766386830",
        "0.900253369"
      ],
      [
        "0.999498548",
        "0.274741780"
      ],
      [
        "-0.548773466",
        "0.189057389"
      ],
      [
        "-0.797898270",
        "0.531547782"
      ],
      [
        "-0.259680717",
        "0.289541380"
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      [
        "0.274150021",
        "-0.616804503"
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    ],
    [
      [
        "-0.051976817",
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        "0.402563489",
        "-0.999498549"
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        "-0.025296930",
        "0.246523273"
      ],
      [
        "-0.574578960",
        "-0.619418960"
      ],
      [
        "0.288145543",
        "0.656582772"
      ],
      [
        "-0.086415454",
        "-0.550480844"
      ]
    ],
    [
      [
        "0.221631775",
        "-0.186776651"
      ],
      [
        "-0.429915508",
        "0.637733016"
      ],
      [
        "0.486456651",
        "0.457596238"
      ],
      [
        "-0.608200474",
        "-0.583014550"
      ],
      [
        "0.999499675",
        "-0.058717775"
      ],
      [
        "0.224188132",
        "0.545339928"
      ]
    ],
    [
      [
        "-0.635477198",
        "0.759666005"
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        "-0.172769285",
        "-0.093099868"
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        "-0.247207931",
        "-0.061222943"
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      [
        "-0.182012953",
        "0.999503375"
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      [
        "-0.062247544",
        "0.208981333"
      ],
      [
        "0.159955704",
        "0.211040819"
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    ],
    [
      [
        "-0.212730803",
        "0.169055791"
      ],
      [
        "-0.088014023",
        "-0.190450117"
      ],
      [
        "0.852554276",
        "-0.908136129"
      ],
      [
        "-0.183129479",
        "-0.690486841"
      ],
      [
        "0.124688320",
        "0.831828316"
      ],
      [
        "-0.999499220",
        "0.333840879"
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    [
      [
        "0.415734154",
        "0.236908204"
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      [
        "-0.812714585",
        "-0.051661127"
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        "0.167856299",
        "0.677738078"
      ],
      [
        "0.452563783",
        "-0.561052799"
      ],
      [
        "-0.999499640",
        "-0.689632974"
      ],
      [
        "-0.393472045",
        "0.529641861"
      ]
    ],
    [
      [
        "0.056507894",
        "0.146961019"
      ],
      [
        "0.535719941",
        "0.437487389"
      ],
      [
        "-0.093995653",
        "0.999500053"
      ],
      [
        "-0.107288592",
        "-0.362163762"
      ],
      [
        "0.820377238",
        "-0.099973500"
      ],
      [
        "-0.597515711",
        "-0.773380866"
      ]
    ]
  ]
}

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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