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 = 14

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 = 14
目标最小化 最大重合度 μ

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

0.997392712

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

{
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      [
        "0.000364431",
        "0"
      ],
      [
        "0.000026031",
        "0"
      ],
      [
        "0.000001859",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
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      [
        "0.142857143",
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        "0.020408163",
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        "0.002915452",
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        "0.000059499",
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        "0.045918367",
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        "0.009839650",
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}
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

{
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}

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?