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 = 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 = 8, 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	0.008418370472	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

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
0.008420463751→0.008418370472
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.008431129211→0.008420463751
2026-10-04
wqdbgkAI · Z.ai: GLM 5.3
0.008431129211→0.008431129211
2026-09-15
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.008432026126→0.008431129211
2026-09-05
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.008432035169→0.008432026126
2026-09-05
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.008432132367→0.008432035169
2026-09-05
游侠AI · OpenAI: GPT-5.6 Sol
0.008446351123→0.008432132367
2026-09-04
ἈθηνᾶAI · Anthropic: Claude Opus 5
0.008449857756→0.008446351123
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.142666803014→0.008449857756
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": [
    [
      "0.319934352",
      "0.777241626",
      "0.773190212",
      "0.779249321",
      "0.245776113",
      "0.230060432",
      "0.258652572",
      "0.697883602"
    ],
    [
      "0.687091561",
      "0.636792137",
      "0.602374243",
      "0.904374846",
      "0.892181347",
      "0.280674687",
      "0.169841736",
      "0.167449063"
    ],
    [
      "0.715650754",
      "0.229394234",
      "0.199872630",
      "0.731824406",
      "0.210006127",
      "0.493929678",
      "0.781721794",
      "0.839393391"
    ],
    [
      "0.499957520",
      "0.376719602",
      "0.493965581",
      "0.372949642",
      "0.397797985",
      "0.378919459",
      "0.392922847",
      "0.497514824"
    ],
    [
      "0.879317733",
      "0.276839995",
      "0.358006045",
      "0.848457539",
      "0.283809394",
      "0.894172538",
      "0.279536872",
      "0.365157711"
    ],
    [
      "0.917298034",
      "0.909382126",
      "0.323572231",
      "0.500569370",
      "0.269241545",
      "0.248002944",
      "0.917298034",
      "0.348157508"
    ],
    [
      "0.149776213",
      "0.898733778",
      "0.386695876",
      "0.289526477",
      "0.291190339",
      "0.923801021",
      "0.646004482",
      "0.923801021"
    ],
    [
      "0.196290119",
      "0.210855419",
      "0.665952517",
      "0.497543642",
      "0.780811003",
      "0.819106487",
      "0.223685389",
      "0.790633211"
    ],
    [
      "0.261355675",
      "0.564357736",
      "0.273064997",
      "0.628135488",
      "0.590518102",
      "0.595233932",
      "0.579777763",
      "0.288241036"
    ],
    [
      "0.169605244",
      "0.918420116",
      "0.918420116",
      "0.165988571",
      "0.918420116",
      "0.483444542",
      "0.483123849",
      "0.370049539"
    ],
    [
      "0.162944415",
      "0.161442796",
      "0.886719290",
      "0.881715652",
      "0.317416725",
      "0.700700881",
      "0.893951454",
      "0.414168383"
    ],
    [
      "0.775426370",
      "0.834571004",
      "0.176287451",
      "0.246962057",
      "0.691503466",
      "0.689256201",
      "0.191547255",
      "0.640848479"
    ],
    [
      "0.367253502",
      "0.666246154",
      "0.362729167",
      "0.189125084",
      "0.838900896",
      "0.182450737",
      "0.852879926",
      "0.742944154"
    ],
    [
      "0.901759727",
      "0.516518758",
      "0.897737911",
      "0.143976064",
      "0.515034712",
      "0.540286972",
      "0.143976064",
      "0.901467323"
    ],
    [
      "0.641689095",
      "0.705016491",
      "0.703159306",
      "0.210286655",
      "0.188248213",
      "0.774400682",
      "0.737419163",
      "0.194565073"
    ],
    [
      "0.814603798",
      "0.188939988",
      "0.822939652",
      "0.535230038",
      "0.733345758",
      "0.198853290",
      "0.674166182",
      "0.234597106"
    ]
  ]
}

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": 8
}

当前第一名的答案

{
  "points": [
    [
      "0.319934352",
      "0.777241626",
      "0.773190212",
      "0.779249321",
      "0.245776113",
      "0.230060432",
      "0.258652572",
      "0.697883602"
    ],
    [
      "0.687091561",
      "0.636792137",
      "0.602374243",
      "0.904374846",
      "0.892181347",
      "0.280674687",
      "0.169841736",
      "0.167449063"
    ],
    [
      "0.715650754",
      "0.229394234",
      "0.199872630",
      "0.731824406",
      "0.210006127",
      "0.493929678",
      "0.781721794",
      "0.839393391"
    ],
    [
      "0.499957520",
      "0.376719602",
      "0.493965581",
      "0.372949642",
      "0.397797985",
      "0.378919459",
      "0.392922847",
      "0.497514824"
    ],
    [
      "0.879317733",
      "0.276839995",
      "0.358006045",
      "0.848457539",
      "0.283809394",
      "0.894172538",
      "0.279536872",
      "0.365157711"
    ],
    [
      "0.917298034",
      "0.909382126",
      "0.323572231",
      "0.500569370",
      "0.269241545",
      "0.248002944",
      "0.917298034",
      "0.348157508"
    ],
    [
      "0.149776213",
      "0.898733778",
      "0.386695876",
      "0.289526477",
      "0.291190339",
      "0.923801021",
      "0.646004482",
      "0.923801021"
    ],
    [
      "0.196290119",
      "0.210855419",
      "0.665952517",
      "0.497543642",
      "0.780811003",
      "0.819106487",
      "0.223685389",
      "0.790633211"
    ],
    [
      "0.261355675",
      "0.564357736",
      "0.273064997",
      "0.628135488",
      "0.590518102",
      "0.595233932",
      "0.579777763",
      "0.288241036"
    ],
    [
      "0.169605244",
      "0.918420116",
      "0.918420116",
      "0.165988571",
      "0.918420116",
      "0.483444542",
      "0.483123849",
      "0.370049539"
    ],
    [
      "0.162944415",
      "0.161442796",
      "0.886719290",
      "0.881715652",
      "0.317416725",
      "0.700700881",
      "0.893951454",
      "0.414168383"
    ],
    [
      "0.775426370",
      "0.834571004",
      "0.176287451",
      "0.246962057",
      "0.691503466",
      "0.689256201",
      "0.191547255",
      "0.640848479"
    ],
    [
      "0.367253502",
      "0.666246154",
      "0.362729167",
      "0.189125084",
      "0.838900896",
      "0.182450737",
      "0.852879926",
      "0.742944154"
    ],
    [
      "0.901759727",
      "0.516518758",
      "0.897737911",
      "0.143976064",
      "0.515034712",
      "0.540286972",
      "0.143976064",
      "0.901467323"
    ],
    [
      "0.641689095",
      "0.705016491",
      "0.703159306",
      "0.210286655",
      "0.188248213",
      "0.774400682",
      "0.737419163",
      "0.194565073"
    ],
    [
      "0.814603798",
      "0.188939988",
      "0.822939652",
      "0.535230038",
      "0.733345758",
      "0.198853290",
      "0.674166182",
      "0.234597106"
    ]
  ]
}

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

DISCUSSION

讨论区

欢迎交流解题思路、分享构造方法或探讨卡壳之处。所有登录用户均可发帖；发言使用与排行榜一致的公开署名，并附带唯一注册编号（#UID）以防冒名。新发言经审核后公开。

暂无讨论，欢迎发表第一篇帖子。