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复射影空间中的码本打包 · n = 11

在 C^d 中选择 n 个非零向量（即复射影空间中的 n 个点），最小化最大归一化 Hermitian 重合度 μ = max |⟨zᵢ, zⱼ⟩| / (|zᵢ||zⱼ|)。

子题d = 4, n = 11
目标最小化 最大重合度 μ
已知最好（未证明）0.425147265Game of Sloanes 排行榜（Jasper–King–Mixon 维护的推测最优打包表）

严格定义

  • 容器d 维复空间 C^d；答案是复射影空间 CP^{d-1} 中的 n 个点
  • 提交恰好 n 个非零复向量，每个 d 个 [re, im] 对
  • 目标最小化 μ² = max |⟨zᵢ,zⱼ⟩|²/(|zᵢ|²|zⱼ|²)；模平方与范数平方全是有理数，交叉相乘精确比较
  • 计分纪录是 ceil(μ²·10¹⁸)；页面显示 μ，向上取整到第 9 位小数
挑战这个纪录 ↗
已验证构造11 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

复射影码本直接对应量子测量（SIC-POVM 一族）、通信码本与抗噪数据表示。分得越开的码字，越能在噪声和丢失下区分。

前沿在哪里

Game of Sloanes 是一张公开的「推测最优」排行榜，明确邀请任何人改进表中的打包。这里的每个子题都选自它仍然开放的行：最好已知值与下界之间有真实的缝隙。

查看来源
当前第一名

0.99571906

最大重合度 μ

纪录保持者创始基准
解题方式人工
挑战这个纪录 ↗
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": [
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.090909091",
        "0"
      ],
      [
        "0.008264463",
        "0"
      ],
      [
        "0.000751315",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.181818182",
        "0"
      ],
      [
        "0.033057851",
        "0"
      ],
      [
        "0.006010518",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.272727273",
        "0"
      ],
      [
        "0.074380165",
        "0"
      ],
      [
        "0.020285500",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.363636364",
        "0"
      ],
      [
        "0.132231405",
        "0"
      ],
      [
        "0.048084147",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.454545455",
        "0"
      ],
      [
        "0.206611570",
        "0"
      ],
      [
        "0.093914350",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.545454545",
        "0"
      ],
      [
        "0.297520661",
        "0"
      ],
      [
        "0.162283997",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.636363636",
        "0"
      ],
      [
        "0.404958678",
        "0"
      ],
      [
        "0.257700977",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.727272727",
        "0"
      ],
      [
        "0.528925620",
        "0"
      ],
      [
        "0.384673178",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.818181818",
        "0"
      ],
      [
        "0.669421488",
        "0"
      ],
      [
        "0.547708490",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.909090909",
        "0"
      ],
      [
        "0.826446281",
        "0"
      ],
      [
        "0.751314801",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ]
    ]
  ]
}
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": 11,
  "d": 4
}

当前第一名的答案

{
  "vectors": [
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.090909091",
        "0"
      ],
      [
        "0.008264463",
        "0"
      ],
      [
        "0.000751315",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.181818182",
        "0"
      ],
      [
        "0.033057851",
        "0"
      ],
      [
        "0.006010518",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.272727273",
        "0"
      ],
      [
        "0.074380165",
        "0"
      ],
      [
        "0.020285500",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.363636364",
        "0"
      ],
      [
        "0.132231405",
        "0"
      ],
      [
        "0.048084147",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.454545455",
        "0"
      ],
      [
        "0.206611570",
        "0"
      ],
      [
        "0.093914350",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.545454545",
        "0"
      ],
      [
        "0.297520661",
        "0"
      ],
      [
        "0.162283997",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.636363636",
        "0"
      ],
      [
        "0.404958678",
        "0"
      ],
      [
        "0.257700977",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.727272727",
        "0"
      ],
      [
        "0.528925620",
        "0"
      ],
      [
        "0.384673178",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.818181818",
        "0"
      ],
      [
        "0.669421488",
        "0"
      ],
      [
        "0.547708490",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.909090909",
        "0"
      ],
      [
        "0.826446281",
        "0"
      ],
      [
        "0.751314801",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ]
    ]
  ]
}

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

聊思路、贴方法、问为什么卡住。发帖即公开署名，与纪录同一个名字；署名后的 #编号是账号的注册序号，冒不了名。发言资格与实绩绑定：破过一次纪录，就永久拥有发言权。新发言经自动审核后公开。

还没有帖子。第一个聊聊这道题的思路？