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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 · n = 10

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 = 6, n = 10
目标最小化 最大重合度 μ
已知最好（未证明）0.272203145Game 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10 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

0.994780215

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"
      ],
      [
        "0.100000000",
        "0"
      ],
      [
        "0.010000000",
        "0"
      ],
      [
        "0.001000000",
        "0"
      ],
      [
        "0.000100000",
        "0"
      ],
      [
        "0.000010000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.200000000",
        "0"
      ],
      [
        "0.040000000",
        "0"
      ],
      [
        "0.008000000",
        "0"
      ],
      [
        "0.001600000",
        "0"
      ],
      [
        "0.000320000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.300000000",
        "0"
      ],
      [
        "0.090000000",
        "0"
      ],
      [
        "0.027000000",
        "0"
      ],
      [
        "0.008100000",
        "0"
      ],
      [
        "0.002430000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.400000000",
        "0"
      ],
      [
        "0.160000000",
        "0"
      ],
      [
        "0.064000000",
        "0"
      ],
      [
        "0.025600000",
        "0"
      ],
      [
        "0.010240000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.500000000",
        "0"
      ],
      [
        "0.250000000",
        "0"
      ],
      [
        "0.125000000",
        "0"
      ],
      [
        "0.062500000",
        "0"
      ],
      [
        "0.031250000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.600000000",
        "0"
      ],
      [
        "0.360000000",
        "0"
      ],
      [
        "0.216000000",
        "0"
      ],
      [
        "0.129600000",
        "0"
      ],
      [
        "0.077760000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.700000000",
        "0"
      ],
      [
        "0.490000000",
        "0"
      ],
      [
        "0.343000000",
        "0"
      ],
      [
        "0.240100000",
        "0"
      ],
      [
        "0.168070000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.800000000",
        "0"
      ],
      [
        "0.640000000",
        "0"
      ],
      [
        "0.512000000",
        "0"
      ],
      [
        "0.409600000",
        "0"
      ],
      [
        "0.327680000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.900000000",
        "0"
      ],
      [
        "0.810000000",
        "0"
      ],
      [
        "0.729000000",
        "0"
      ],
      [
        "0.656100000",
        "0"
      ],
      [
        "0.590490000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "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": 10,
  "d": 6
}

当前第一名的答案

{
  "vectors": [
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.100000000",
        "0"
      ],
      [
        "0.010000000",
        "0"
      ],
      [
        "0.001000000",
        "0"
      ],
      [
        "0.000100000",
        "0"
      ],
      [
        "0.000010000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.200000000",
        "0"
      ],
      [
        "0.040000000",
        "0"
      ],
      [
        "0.008000000",
        "0"
      ],
      [
        "0.001600000",
        "0"
      ],
      [
        "0.000320000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.300000000",
        "0"
      ],
      [
        "0.090000000",
        "0"
      ],
      [
        "0.027000000",
        "0"
      ],
      [
        "0.008100000",
        "0"
      ],
      [
        "0.002430000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.400000000",
        "0"
      ],
      [
        "0.160000000",
        "0"
      ],
      [
        "0.064000000",
        "0"
      ],
      [
        "0.025600000",
        "0"
      ],
      [
        "0.010240000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.500000000",
        "0"
      ],
      [
        "0.250000000",
        "0"
      ],
      [
        "0.125000000",
        "0"
      ],
      [
        "0.062500000",
        "0"
      ],
      [
        "0.031250000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.600000000",
        "0"
      ],
      [
        "0.360000000",
        "0"
      ],
      [
        "0.216000000",
        "0"
      ],
      [
        "0.129600000",
        "0"
      ],
      [
        "0.077760000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.700000000",
        "0"
      ],
      [
        "0.490000000",
        "0"
      ],
      [
        "0.343000000",
        "0"
      ],
      [
        "0.240100000",
        "0"
      ],
      [
        "0.168070000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.800000000",
        "0"
      ],
      [
        "0.640000000",
        "0"
      ],
      [
        "0.512000000",
        "0"
      ],
      [
        "0.409600000",
        "0"
      ],
      [
        "0.327680000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.900000000",
        "0"
      ],
      [
        "0.810000000",
        "0"
      ],
      [
        "0.729000000",
        "0"
      ],
      [
        "0.656100000",
        "0"
      ],
      [
        "0.590490000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "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？