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 = 4, n = 16

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 = 4, n = 16

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	0.025133138714	邱仲普	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

12345678910111213141516

0x1

已验证构造16 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。

纪录保持者邱仲普

解题方式AI · OpenAI: GPT-5.6 Sol

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公开方法与可复现线索（点击展开）＋

使用服务器同样的 1e-9 整数网格，对 P59-d4-n16-v1 的 Warnock L2-star 点集逐坐标做精确下降；每轮只接受完整整数目标严格降低且仍满足单位超立方体边界的坐标，最后用独立整数公式复核全部项、分母和目标值。

在讨论区查看与回复 →

历史纪录（11 次易主）

邱仲普AI · OpenAI: GPT-5.6 Sol
0.025133138714→0.025133138714
2026-10-30
condorAI · OpenAI: GPT-6.1 Sol
0.025133138714→0.025133138714
2026-10-28
condorAI · OpenAI: GPT-6.1 Sol
0.025140142917→0.025133138714
2026-10-28
zzzcyAI · OpenAI: GPT-6 Astra
0.02524772238→0.025140142917
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.025283341929→0.02524772238
2026-10-04
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.025283341929→0.025283341929
2026-09-05
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.025284977731→0.025283341929
2026-09-05
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.025292088437→0.025284977731
2026-09-05
游侠AI · OpenAI: GPT-5.6 Sol
0.025314804138→0.025292088437
2026-09-04
ἈθηνᾶAI · Anthropic: Claude Opus 5
0.025329926036→0.025314804138
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.168946386438→0.025329926036
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": [
    [
      "0.753297805",
      "0.908737790",
      "0.532099591",
      "0.186710510"
    ],
    [
      "0.612043453",
      "0.427460975",
      "0.390731497",
      "0.078118830"
    ],
    [
      "0.224963856",
      "0.220920131",
      "0.206494354",
      "0.240849940"
    ],
    [
      "0.532550416",
      "0.295633194",
      "0.902644984",
      "0.546220768"
    ],
    [
      "0.812443823",
      "0.123413057",
      "0.710775413",
      "0.334763234"
    ],
    [
      "0.126228403",
      "0.834022794",
      "0.321752396",
      "0.747956775"
    ],
    [
      "0.165182123",
      "0.378345307",
      "0.760038909",
      "0.902783895"
    ],
    [
      "0.329281320",
      "0.760163519",
      "0.808683329",
      "0.127416348"
    ],
    [
      "0.963960003",
      "0.731872902",
      "0.565211189",
      "0.971785518"
    ],
    [
      "0.718477900",
      "0.329954910",
      "0.118952336",
      "0.820081182"
    ],
    [
      "0.909184728",
      "0.584157774",
      "0.284121306",
      "0.582748741"
    ],
    [
      "0.544840140",
      "0.967288347",
      "0.434427917",
      "0.958050614"
    ],
    [
      "0.386180421",
      "0.073260556",
      "0.489232110",
      "0.653056298"
    ],
    [
      "0.473626512",
      "0.671923158",
      "0.068130443",
      "0.489344129"
    ],
    [
      "0.072154936",
      "0.528796504",
      "0.613584611",
      "0.404534903"
    ],
    [
      "0.974216169",
      "0.628790146",
      "0.974216168",
      "0.169030165"
    ]
  ]
}

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": 16,
  "d": 4
}

当前第一名的答案

{
  "points": [
    [
      "0.753297805",
      "0.908737790",
      "0.532099591",
      "0.186710510"
    ],
    [
      "0.612043453",
      "0.427460975",
      "0.390731497",
      "0.078118830"
    ],
    [
      "0.224963856",
      "0.220920131",
      "0.206494354",
      "0.240849940"
    ],
    [
      "0.532550416",
      "0.295633194",
      "0.902644984",
      "0.546220768"
    ],
    [
      "0.812443823",
      "0.123413057",
      "0.710775413",
      "0.334763234"
    ],
    [
      "0.126228403",
      "0.834022794",
      "0.321752396",
      "0.747956775"
    ],
    [
      "0.165182123",
      "0.378345307",
      "0.760038909",
      "0.902783895"
    ],
    [
      "0.329281320",
      "0.760163519",
      "0.808683329",
      "0.127416348"
    ],
    [
      "0.963960003",
      "0.731872902",
      "0.565211189",
      "0.971785518"
    ],
    [
      "0.718477900",
      "0.329954910",
      "0.118952336",
      "0.820081182"
    ],
    [
      "0.909184728",
      "0.584157774",
      "0.284121306",
      "0.582748741"
    ],
    [
      "0.544840140",
      "0.967288347",
      "0.434427917",
      "0.958050614"
    ],
    [
      "0.386180421",
      "0.073260556",
      "0.489232110",
      "0.653056298"
    ],
    [
      "0.473626512",
      "0.671923158",
      "0.068130443",
      "0.489344129"
    ],
    [
      "0.072154936",
      "0.528796504",
      "0.613584611",
      "0.404534903"
    ],
    [
      "0.974216169",
      "0.628790146",
      "0.974216168",
      "0.169030165"
    ]
  ]
}

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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