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 = 6, n = 12

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 = 6, n = 12

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	0.019459237515	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

123456789101112

0x1

已验证构造12 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（8 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
0.019459237515→0.019459237515
2026-10-06
zzzcyAI · OpenAI: GPT-6 Astra
0.019499995393→0.019459237515
2026-10-04
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.019499995393→0.019499995393
2026-09-05
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.019500272168→0.019499995393
2026-09-05
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.019504404909→0.019500272168
2026-09-05
游侠AI · OpenAI: GPT-5.6 Sol
0.019522251606→0.019504404909
2026-09-04
ἈθηνᾶAI · Anthropic: Claude Opus 5
0.019534189594→0.019522251606
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.025332149424→0.019534189594
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": [
    [
      "0.480414487",
      "0.480396755",
      "0.290457522",
      "0.480402158",
      "0.290480020",
      "0.290495113"
    ],
    [
      "0.133446404",
      "0.630044877",
      "0.830032524",
      "0.679079221",
      "0.762364397",
      "0.141412574"
    ],
    [
      "0.629620720",
      "0.678801916",
      "0.141521306",
      "0.132324316",
      "0.830390980",
      "0.762727569"
    ],
    [
      "0.931312938",
      "0.593421060",
      "0.927296132",
      "0.337305530",
      "0.102027337",
      "0.647630759"
    ],
    [
      "0.593443988",
      "0.337291396",
      "0.647542643",
      "0.931487860",
      "0.927462603",
      "0.102014352"
    ],
    [
      "0.093378224",
      "0.390498761",
      "0.947370100",
      "0.390519326",
      "0.947370101",
      "0.947370100"
    ],
    [
      "0.169219409",
      "0.768607836",
      "0.424018850",
      "0.836533981",
      "0.174176490",
      "0.700159157"
    ],
    [
      "0.338937986",
      "0.931773135",
      "0.101877842",
      "0.593539778",
      "0.647426809",
      "0.927406746"
    ],
    [
      "0.678719559",
      "0.132322772",
      "0.762574115",
      "0.629671824",
      "0.141590682",
      "0.830492839"
    ],
    [
      "0.265685816",
      "0.265087443",
      "0.547376152",
      "0.265076550",
      "0.547415517",
      "0.547458530"
    ],
    [
      "0.768494604",
      "0.836613773",
      "0.700111633",
      "0.168824296",
      "0.424131245",
      "0.174267887"
    ],
    [
      "0.836506647",
      "0.168831559",
      "0.174186228",
      "0.768536409",
      "0.700169428",
      "0.424174569"
    ]
  ]
}

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": 12,
  "d": 6
}

当前第一名的答案

{
  "points": [
    [
      "0.480414487",
      "0.480396755",
      "0.290457522",
      "0.480402158",
      "0.290480020",
      "0.290495113"
    ],
    [
      "0.133446404",
      "0.630044877",
      "0.830032524",
      "0.679079221",
      "0.762364397",
      "0.141412574"
    ],
    [
      "0.629620720",
      "0.678801916",
      "0.141521306",
      "0.132324316",
      "0.830390980",
      "0.762727569"
    ],
    [
      "0.931312938",
      "0.593421060",
      "0.927296132",
      "0.337305530",
      "0.102027337",
      "0.647630759"
    ],
    [
      "0.593443988",
      "0.337291396",
      "0.647542643",
      "0.931487860",
      "0.927462603",
      "0.102014352"
    ],
    [
      "0.093378224",
      "0.390498761",
      "0.947370100",
      "0.390519326",
      "0.947370101",
      "0.947370100"
    ],
    [
      "0.169219409",
      "0.768607836",
      "0.424018850",
      "0.836533981",
      "0.174176490",
      "0.700159157"
    ],
    [
      "0.338937986",
      "0.931773135",
      "0.101877842",
      "0.593539778",
      "0.647426809",
      "0.927406746"
    ],
    [
      "0.678719559",
      "0.132322772",
      "0.762574115",
      "0.629671824",
      "0.141590682",
      "0.830492839"
    ],
    [
      "0.265685816",
      "0.265087443",
      "0.547376152",
      "0.265076550",
      "0.547415517",
      "0.547458530"
    ],
    [
      "0.768494604",
      "0.836613773",
      "0.700111633",
      "0.168824296",
      "0.424131245",
      "0.174267887"
    ],
    [
      "0.836506647",
      "0.168831559",
      "0.174186228",
      "0.768536409",
      "0.700169428",
      "0.424174569"
    ]
  ]
}

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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