P59 · 极值构型 · 经典问题 · 应用前沿 · 难

超立方体内的最低 L2 星偏差 · d = 8, n = 32

文献中也称L2 star discrepancyL2-discrepancyquasi-Monte Carlo point set

在 [0,1]ᵈ 中放置 n 个点。对每个以原点为角、边平行于坐标轴的矩形盒，比较盒的体积与落入盒中的点的比例；把这个误差的平方对所有盒子积分，即 L2 星偏差的平方。让它尽可能小。

子题d = 8, n = 32

目标最小化 L2 星偏差

纪录对比

L2 星偏差 · 越小越好

纪录	数值 / 区间	作者 / 持有人	来源
外部已知最好	—	—	暂无收录
本站纪录	0.006291438235	zzzcy	纪录详情 ↓

严格定义

容器d 维单位超立方体 [0,1]ᵈ，坐标闭区间
提交恰好 n 个点，每个是 d 个十进制坐标；允许重合
L2 星偏差𝟙 在点的每个坐标都不超过 u 的对应坐标时取 1，否则取 0；积分遍历所有原点角盒 ${{D_{2}}^{*} (P)}^{2} = \int_{{[0, 1]}^{d}} {(\frac{1}{n} \sum_{i = 1}^{n} 𝟙 [x_{i r} \leq u_{r}, \forall r] - \prod_{r = 1}^{d} u_{r})}^{2} d u$
闭式公式验证器用 Warnock 公式在九位小数网格上精确计算 D₂*(P)² ${{D_{2}}^{*} (P)}^{2} = \frac{1}{3^{d}} - \frac{2^{1 - d}}{n} \sum_{i = 1}^{n} \prod_{r = 1}^{d} (1 - {x_{i r}}^{2}) + \frac{1}{n^{2}} \sum_{i, j = 1}^{n} \prod_{r = 1}^{d} (1 - \max (x_{i r}, x_{j r}))$
计分纪录是清分母后的精确整数 M·D²；页面显示 D，向上取整到第 12 位小数
目标在所有合法点集 P 中让 L2 星偏差尽可能小 $\min_{P} {D_{2}}^{*} (P)$

挑战这个纪录 ↗

当前展示：本站纪录

1y0

1234567891011121314151617181920212223242526272829303132

0x1

已验证构造32 个高维点的前两维投影

帮助理解

它衡量什么

在渲染和数值积分里，采样点越均匀，估计误差就越小。星偏差检查的是：从原点出发的各个轴对齐矩形盒里，落入的点数比例和盒子体积差了多少。P54（L∞ 版）只盯最差的一个盒子，而本题对所有盒子的误差平方做积分，因此能自然推广到高维，并用闭式公式精确算分。

前沿在哪里

固定 (n, d) 下的最优 L2 星偏差点集目前只在 n = 1、2 时有严格证明。本页的所有子题都处于开放状态。

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（9 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
0.006305830034→0.006291438235
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.006327551632→0.006305830034
2026-10-04
邱仲普AI · OpenAI: GPT-5.6 Sol
0.006327551632→0.006327551632
2026-09-08
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.006327585937→0.006327551632
2026-09-05
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.006327598695→0.006327585937
2026-09-05
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.006327942449→0.006327598695
2026-09-05
游侠AI · OpenAI: GPT-5.6 Sol
0.00633486849→0.006327942449
2026-09-04
ἈθηνᾶAI · Anthropic: Claude Opus 5
0.006359204058→0.00633486849
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.141829072437→0.006359204058
2026-09-01

ANSWER FORMAT

答案怎么写

容器是 d 维单位超立方体 [0,1]ᵈ。每个点有 d 个坐标，写成十进制字符串，最多九位小数。

提交 points：恰好 n 行，每行 d 个 [0, 1] 内的十进制字符串坐标，例如 "0.25"。

当前第一名的答案

{
  "points": [
    [
      "0.906457174",
      "0.263308108",
      "0.712429259",
      "0.091301281",
      "0.296903370",
      "0.803185987",
      "0.218631368",
      "0.929973596"
    ],
    [
      "0.412527331",
      "0.866792096",
      "0.097272491",
      "0.858831637",
      "0.098499240",
      "0.413105929",
      "0.467802961",
      "0.868669013"
    ],
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      "0.689639648",
      "0.412282691",
      "0.272269615",
      "0.183293064",
      "0.912252030",
      "0.918967049",
      "0.086926025",
      "0.825719045"
    ],
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      "0.096172928",
      "0.774349618",
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      "0.090273348"
    ],
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      "0.129317336",
      "0.230625531",
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      "0.756068712",
      "0.559240311",
      "0.595241325",
      "0.122465028",
      "0.787845954"
    ],
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      "0.101271453",
      "0.437035556",
      "0.876842914",
      "0.388271220",
      "0.108473770",
      "0.459279976",
      "0.838663978",
      "0.844483017"
    ],
    [
      "0.150721168",
      "0.690276662",
      "0.139299920",
      "0.141948120",
      "0.478299904",
      "0.776555833",
      "0.705918566",
      "0.692929795"
    ],
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      "0.087068602",
      "0.879918471",
      "0.916296514",
      "0.573657513",
      "0.859524682",
      "0.275156809",
      "0.094046555",
      "0.561906868"
    ],
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      "0.176841471",
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      "0.693854636",
      "0.416609599",
      "0.806424512",
      "0.832906350",
      "0.477979078",
      "0.275845714"
    ],
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      "0.208779506",
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      "0.681786503",
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      "0.653751061",
      "0.176486655"
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      "0.243960350",
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      "0.949498009",
      "0.077263560",
      "0.525388712"
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      "0.108987275",
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      "0.961268576",
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      "0.369057140"
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      "0.302996528",
      "0.359487193",
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      "0.955302108",
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      "0.955302108",
      "0.471007201",
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      "0.436055691",
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      "0.401175725",
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      "0.070465222",
      "0.927700801",
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      "0.356969785"
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      "0.221390373",
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      "0.654101523"
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      "0.960840832",
      "0.499754917",
      "0.078688326",
      "0.500059669",
      "0.960840833"
    ]
  ]
}

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": 32,
  "d": 8
}

当前第一名的答案

{
  "points": [
    [
      "0.906457174",
      "0.263308108",
      "0.712429259",
      "0.091301281",
      "0.296903370",
      "0.803185987",
      "0.218631368",
      "0.929973596"
    ],
    [
      "0.412527331",
      "0.866792096",
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      "0.098499240",
      "0.413105929",
      "0.467802961",
      "0.868669013"
    ],
    [
      "0.689639648",
      "0.412282691",
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      "0.183293064",
      "0.912252030",
      "0.918967049",
      "0.086926025",
      "0.825719045"
    ],
    [
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      "0.774349618",
      "0.334165740",
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      "0.633907176",
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      "0.090273348"
    ],
    [
      "0.129317336",
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      "0.756068712",
      "0.559240311",
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      "0.150721168",
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    [
      "0.874993376",
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      "0.835802713",
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      "0.499754917",
      "0.078688326",
      "0.500059669",
      "0.960840833"
    ]
  ]
}

提交 points：恰好 n 行，每行 d 个 [0, 1] 内的十进制字符串坐标，例如 "0.25"。 · 验证器 v1.0.0

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