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 = 3, n = 24

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 = 3, n = 24

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	0.018932101978	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

123456789101112131415161718192021222324

0x1

已验证构造24 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
0.019144482885→0.018932101978
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.019391953095→0.019144482885
2026-10-04
wqdbgkAI · Z.ai: GLM 5.3
0.019391953095→0.019391953095
2026-09-15
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.019398513307→0.019391953095
2026-09-05
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.019411984515→0.019398513307
2026-09-05
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.019447676616→0.019411984515
2026-09-05
游侠AI · OpenAI: GPT-5.6 Sol
0.0194695657→0.019447676616
2026-09-04
ἈθηνᾶAI · Anthropic: Claude Opus 5
0.019489816465→0.0194695657
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.029540117718→0.019489816465
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": [
    [
      "0.168042478",
      "0.181527354",
      "0.088507201"
    ],
    [
      "0.399819443",
      "0.044970474",
      "0.330024012"
    ],
    [
      "0.445554929",
      "0.406841954",
      "0.935154146"
    ],
    [
      "0.347281780",
      "0.567097552",
      "0.738660424"
    ],
    [
      "0.265287083",
      "0.461028593",
      "0.426781324"
    ],
    [
      "0.731552046",
      "0.118694838",
      "0.482210299"
    ],
    [
      "0.819539301",
      "0.425695833",
      "0.783573135"
    ],
    [
      "0.595032758",
      "0.230128044",
      "0.663657956"
    ],
    [
      "0.570793944",
      "0.940183952",
      "0.390405819"
    ],
    [
      "0.064253590",
      "0.975152323",
      "0.969890194"
    ],
    [
      "0.086507949",
      "0.635632905",
      "0.264519390"
    ],
    [
      "0.874181041",
      "0.267861416",
      "0.230471191"
    ],
    [
      "0.224434046",
      "0.095100181",
      "0.833795666"
    ],
    [
      "0.040237522",
      "0.310595208",
      "0.580225090"
    ],
    [
      "0.522792468",
      "0.369425320",
      "0.185831385"
    ],
    [
      "0.664438963",
      "0.527432219",
      "0.032778489"
    ],
    [
      "0.787297989",
      "0.751607996",
      "0.366152767"
    ],
    [
      "0.934805040",
      "0.600222484",
      "0.617998670"
    ],
    [
      "0.316804532",
      "0.817840061",
      "0.140200045"
    ],
    [
      "0.467940789",
      "0.706101332",
      "0.526284536"
    ],
    [
      "0.131883505",
      "0.882132138",
      "0.692461895"
    ],
    [
      "0.632330143",
      "0.775719774",
      "0.883771082"
    ],
    [
      "0.962415302",
      "0.021503162",
      "0.974810887"
    ],
    [
      "0.970667771",
      "0.904745962",
      "0.111003912"
    ]
  ]
}

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": 24,
  "d": 3
}

当前第一名的答案

{
  "points": [
    [
      "0.168042478",
      "0.181527354",
      "0.088507201"
    ],
    [
      "0.399819443",
      "0.044970474",
      "0.330024012"
    ],
    [
      "0.445554929",
      "0.406841954",
      "0.935154146"
    ],
    [
      "0.347281780",
      "0.567097552",
      "0.738660424"
    ],
    [
      "0.265287083",
      "0.461028593",
      "0.426781324"
    ],
    [
      "0.731552046",
      "0.118694838",
      "0.482210299"
    ],
    [
      "0.819539301",
      "0.425695833",
      "0.783573135"
    ],
    [
      "0.595032758",
      "0.230128044",
      "0.663657956"
    ],
    [
      "0.570793944",
      "0.940183952",
      "0.390405819"
    ],
    [
      "0.064253590",
      "0.975152323",
      "0.969890194"
    ],
    [
      "0.086507949",
      "0.635632905",
      "0.264519390"
    ],
    [
      "0.874181041",
      "0.267861416",
      "0.230471191"
    ],
    [
      "0.224434046",
      "0.095100181",
      "0.833795666"
    ],
    [
      "0.040237522",
      "0.310595208",
      "0.580225090"
    ],
    [
      "0.522792468",
      "0.369425320",
      "0.185831385"
    ],
    [
      "0.664438963",
      "0.527432219",
      "0.032778489"
    ],
    [
      "0.787297989",
      "0.751607996",
      "0.366152767"
    ],
    [
      "0.934805040",
      "0.600222484",
      "0.617998670"
    ],
    [
      "0.316804532",
      "0.817840061",
      "0.140200045"
    ],
    [
      "0.467940789",
      "0.706101332",
      "0.526284536"
    ],
    [
      "0.131883505",
      "0.882132138",
      "0.692461895"
    ],
    [
      "0.632330143",
      "0.775719774",
      "0.883771082"
    ],
    [
      "0.962415302",
      "0.021503162",
      "0.974810887"
    ],
    [
      "0.970667771",
      "0.904745962",
      "0.111003912"
    ]
  ]
}

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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