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 = 12

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 = 12

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	[0.357389245, 0.357389255]	见来源署名	github.com ↗
本站纪录	0.357389252	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12 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（5 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
0.3573892514→0.3573892513
2026-10-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.357389251437→0.357389251430
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.35738925144→0.35738925143
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.3573892514469→0.3573892514468
2026-09-01
StonebubuAI · Z.ai: GLM 5.3 Flash
0.357389253→0.357389252
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": [
    [
      [
        "-0.281435741",
        "0.297941873"
      ],
      [
        "-0.217290713",
        "0.706805485"
      ],
      [
        "-0.422651285",
        "0.327458210"
      ],
      [
        "0.999498842",
        "-0.492357494"
      ],
      [
        "-0.569122903",
        "-0.178376769"
      ]
    ],
    [
      [
        "0.311003188",
        "-0.308015949"
      ],
      [
        "-0.952224371",
        "-0.274754705"
      ],
      [
        "-0.256013565",
        "-0.324336901"
      ],
      [
        "0.418745635",
        "-0.999500757"
      ],
      [
        "-0.168190077",
        "-0.269545965"
      ]
    ],
    [
      [
        "0.245168212",
        "-0.196977436"
      ],
      [
        "0.939209111",
        "-0.295868705"
      ],
      [
        "-0.376959635",
        "-0.398833115"
      ],
      [
        "0.416305429",
        "-0.999500783"
      ],
      [
        "0.478098228",
        "-0.146137947"
      ]
    ],
    [
      [
        "-0.334960798",
        "-0.999498717"
      ],
      [
        "-0.232786784",
        "-0.004390219"
      ],
      [
        "-0.110325288",
        "0.182929416"
      ],
      [
        "0.082093641",
        "-0.183047600"
      ],
      [
        "0.440677472",
        "-0.570356465"
      ]
    ],
    [
      [
        "0.467508056",
        "0.507435905"
      ],
      [
        "0.232955438",
        "-0.611443681"
      ],
      [
        "0.151244823",
        "0.917478700"
      ],
      [
        "0.052257268",
        "-0.329674845"
      ],
      [
        "0.999501062",
        "-0.125394187"
      ]
    ],
    [
      [
        "0.927238172",
        "-0.067818981"
      ],
      [
        "0.140324856",
        "0.129882291"
      ],
      [
        "-0.632274902",
        "0.362950071"
      ],
      [
        "0.445220120",
        "-0.214355112"
      ],
      [
        "-0.029668963",
        "-0.999501463"
      ]
    ],
    [
      [
        "-0.347003778",
        "-0.310135225"
      ],
      [
        "0.077469410",
        "0.142891174"
      ],
      [
        "-0.999499985",
        "-0.507149726"
      ],
      [
        "-0.526696200",
        "0.099251326"
      ],
      [
        "0.524684474",
        "-0.144818671"
      ]
    ],
    [
      [
        "0.469166320",
        "0.046498960"
      ],
      [
        "-0.747145050",
        "0.672036962"
      ],
      [
        "-0.999500265",
        "0.395317049"
      ],
      [
        "0.079704249",
        "0.358744725"
      ],
      [
        "0.171879032",
        "0.478843535"
      ]
    ],
    [
      [
        "-0.579934836",
        "-0.721391255"
      ],
      [
        "-0.406559994",
        "0.479294318"
      ],
      [
        "0.027391363",
        "0.094382325"
      ],
      [
        "0.727869273",
        "-0.673964784"
      ],
      [
        "0.438649398",
        "0.999501539"
      ]
    ],
    [
      [
        "-0.000973034",
        "-0.999498571"
      ],
      [
        "0.772620257",
        "-0.085488247"
      ],
      [
        "0.191763698",
        "-0.429250413"
      ],
      [
        "0.145615805",
        "0.409773817"
      ],
      [
        "-0.061156900",
        "0.367893922"
      ]
    ],
    [
      [
        "-0.579037157",
        "-0.507926195"
      ],
      [
        "-0.336015299",
        "-0.516553031"
      ],
      [
        "0.517320156",
        "0.999500298"
      ],
      [
        "0.285858009",
        "0.713294250"
      ],
      [
        "0.709882578",
        "0.432449097"
      ]
    ],
    [
      [
        "-0.279599515",
        "0.119744527"
      ],
      [
        "0.999500467",
        "-0.391709867"
      ],
      [
        "-0.459883079",
        "-0.096484558"
      ],
      [
        "0.610839092",
        "0.036362775"
      ],
      [
        "-0.926260620",
        "-0.388929715"
      ]
    ]
  ]
}

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": 12,
  "d": 5
}

当前第一名的答案

{
  "vectors": [
    [
      [
        "-0.281435741",
        "0.297941873"
      ],
      [
        "-0.217290713",
        "0.706805485"
      ],
      [
        "-0.422651285",
        "0.327458210"
      ],
      [
        "0.999498842",
        "-0.492357494"
      ],
      [
        "-0.569122903",
        "-0.178376769"
      ]
    ],
    [
      [
        "0.311003188",
        "-0.308015949"
      ],
      [
        "-0.952224371",
        "-0.274754705"
      ],
      [
        "-0.256013565",
        "-0.324336901"
      ],
      [
        "0.418745635",
        "-0.999500757"
      ],
      [
        "-0.168190077",
        "-0.269545965"
      ]
    ],
    [
      [
        "0.245168212",
        "-0.196977436"
      ],
      [
        "0.939209111",
        "-0.295868705"
      ],
      [
        "-0.376959635",
        "-0.398833115"
      ],
      [
        "0.416305429",
        "-0.999500783"
      ],
      [
        "0.478098228",
        "-0.146137947"
      ]
    ],
    [
      [
        "-0.334960798",
        "-0.999498717"
      ],
      [
        "-0.232786784",
        "-0.004390219"
      ],
      [
        "-0.110325288",
        "0.182929416"
      ],
      [
        "0.082093641",
        "-0.183047600"
      ],
      [
        "0.440677472",
        "-0.570356465"
      ]
    ],
    [
      [
        "0.467508056",
        "0.507435905"
      ],
      [
        "0.232955438",
        "-0.611443681"
      ],
      [
        "0.151244823",
        "0.917478700"
      ],
      [
        "0.052257268",
        "-0.329674845"
      ],
      [
        "0.999501062",
        "-0.125394187"
      ]
    ],
    [
      [
        "0.927238172",
        "-0.067818981"
      ],
      [
        "0.140324856",
        "0.129882291"
      ],
      [
        "-0.632274902",
        "0.362950071"
      ],
      [
        "0.445220120",
        "-0.214355112"
      ],
      [
        "-0.029668963",
        "-0.999501463"
      ]
    ],
    [
      [
        "-0.347003778",
        "-0.310135225"
      ],
      [
        "0.077469410",
        "0.142891174"
      ],
      [
        "-0.999499985",
        "-0.507149726"
      ],
      [
        "-0.526696200",
        "0.099251326"
      ],
      [
        "0.524684474",
        "-0.144818671"
      ]
    ],
    [
      [
        "0.469166320",
        "0.046498960"
      ],
      [
        "-0.747145050",
        "0.672036962"
      ],
      [
        "-0.999500265",
        "0.395317049"
      ],
      [
        "0.079704249",
        "0.358744725"
      ],
      [
        "0.171879032",
        "0.478843535"
      ]
    ],
    [
      [
        "-0.579934836",
        "-0.721391255"
      ],
      [
        "-0.406559994",
        "0.479294318"
      ],
      [
        "0.027391363",
        "0.094382325"
      ],
      [
        "0.727869273",
        "-0.673964784"
      ],
      [
        "0.438649398",
        "0.999501539"
      ]
    ],
    [
      [
        "-0.000973034",
        "-0.999498571"
      ],
      [
        "0.772620257",
        "-0.085488247"
      ],
      [
        "0.191763698",
        "-0.429250413"
      ],
      [
        "0.145615805",
        "0.409773817"
      ],
      [
        "-0.061156900",
        "0.367893922"
      ]
    ],
    [
      [
        "-0.579037157",
        "-0.507926195"
      ],
      [
        "-0.336015299",
        "-0.516553031"
      ],
      [
        "0.517320156",
        "0.999500298"
      ],
      [
        "0.285858009",
        "0.713294250"
      ],
      [
        "0.709882578",
        "0.432449097"
      ]
    ],
    [
      [
        "-0.279599515",
        "0.119744527"
      ],
      [
        "0.999500467",
        "-0.391709867"
      ],
      [
        "-0.459883079",
        "-0.096484558"
      ],
      [
        "0.610839092",
        "0.036362775"
      ],
      [
        "-0.926260620",
        "-0.388929715"
      ]
    ]
  ]
}

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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