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 = 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 = 6, 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	0.016684273988	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（8 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
0.016684273988→0.016684273988
2026-10-06
zzzcyAI · OpenAI: GPT-6 Astra
0.016716576154→0.016684273988
2026-10-04
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.016716617167→0.016716576154
2026-09-05
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.016716617167→0.016716617167
2026-09-05
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.016732247762→0.016716617167
2026-09-05
游侠AI · OpenAI: GPT-5.6 Sol
0.016770732444→0.016732247762
2026-09-04
ἈθηνᾶAI · Anthropic: Claude Opus 5
0.016793584418→0.016770732444
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.020761999922→0.016793584418
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": [
    [
      "0.077281922",
      "0.931163590",
      "0.347517830",
      "0.885366290",
      "0.917916951",
      "0.375127493"
    ],
    [
      "0.120507231",
      "0.135195098",
      "0.604516486",
      "0.729736944",
      "0.620784997",
      "0.672710514"
    ],
    [
      "0.253944517",
      "0.545873620",
      "0.227643208",
      "0.502132837",
      "0.237130540",
      "0.501623882"
    ],
    [
      "0.084896475",
      "0.911297936",
      "0.895045604",
      "0.319469067",
      "0.330713801",
      "0.911547652"
    ],
    [
      "0.459677181",
      "0.250201985",
      "0.427998596",
      "0.257361243",
      "0.450529070",
      "0.286929853"
    ],
    [
      "0.570536405",
      "0.326011997",
      "0.108905146",
      "0.904098057",
      "0.363637243",
      "0.892743511"
    ],
    [
      "0.685523005",
      "0.425481281",
      "0.700728026",
      "0.133847691",
      "0.143595378",
      "0.755876951"
    ],
    [
      "0.346894723",
      "0.744175210",
      "0.318374377",
      "0.094601363",
      "0.861275975",
      "0.833704368"
    ],
    [
      "0.944504930",
      "0.089255153",
      "0.297603198",
      "0.406338697",
      "0.685806013",
      "0.944504931"
    ],
    [
      "0.745893928",
      "0.383451443",
      "0.144673912",
      "0.653677463",
      "0.794788864",
      "0.178261383"
    ],
    [
      "0.608326680",
      "0.790035744",
      "0.551544145",
      "0.800670850",
      "0.111822711",
      "0.111822711"
    ],
    [
      "0.827986917",
      "0.856740934",
      "0.099415851",
      "0.172037884",
      "0.571648260",
      "0.608877452"
    ],
    [
      "0.884584391",
      "0.107654003",
      "0.857707618",
      "0.603908293",
      "0.299405580",
      "0.402277148"
    ],
    [
      "0.341831734",
      "0.448321773",
      "0.953259543",
      "0.953259543",
      "0.344194313",
      "0.358752345"
    ],
    [
      "0.579548556",
      "0.075367370",
      "0.925653833",
      "0.423408042",
      "0.941589399",
      "0.578405675"
    ],
    [
      "0.170448672",
      "0.688457021",
      "0.782263226",
      "0.368615326",
      "0.706626401",
      "0.145975041"
    ]
  ]
}

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": 6
}

当前第一名的答案

{
  "points": [
    [
      "0.077281922",
      "0.931163590",
      "0.347517830",
      "0.885366290",
      "0.917916951",
      "0.375127493"
    ],
    [
      "0.120507231",
      "0.135195098",
      "0.604516486",
      "0.729736944",
      "0.620784997",
      "0.672710514"
    ],
    [
      "0.253944517",
      "0.545873620",
      "0.227643208",
      "0.502132837",
      "0.237130540",
      "0.501623882"
    ],
    [
      "0.084896475",
      "0.911297936",
      "0.895045604",
      "0.319469067",
      "0.330713801",
      "0.911547652"
    ],
    [
      "0.459677181",
      "0.250201985",
      "0.427998596",
      "0.257361243",
      "0.450529070",
      "0.286929853"
    ],
    [
      "0.570536405",
      "0.326011997",
      "0.108905146",
      "0.904098057",
      "0.363637243",
      "0.892743511"
    ],
    [
      "0.685523005",
      "0.425481281",
      "0.700728026",
      "0.133847691",
      "0.143595378",
      "0.755876951"
    ],
    [
      "0.346894723",
      "0.744175210",
      "0.318374377",
      "0.094601363",
      "0.861275975",
      "0.833704368"
    ],
    [
      "0.944504930",
      "0.089255153",
      "0.297603198",
      "0.406338697",
      "0.685806013",
      "0.944504931"
    ],
    [
      "0.745893928",
      "0.383451443",
      "0.144673912",
      "0.653677463",
      "0.794788864",
      "0.178261383"
    ],
    [
      "0.608326680",
      "0.790035744",
      "0.551544145",
      "0.800670850",
      "0.111822711",
      "0.111822711"
    ],
    [
      "0.827986917",
      "0.856740934",
      "0.099415851",
      "0.172037884",
      "0.571648260",
      "0.608877452"
    ],
    [
      "0.884584391",
      "0.107654003",
      "0.857707618",
      "0.603908293",
      "0.299405580",
      "0.402277148"
    ],
    [
      "0.341831734",
      "0.448321773",
      "0.953259543",
      "0.953259543",
      "0.344194313",
      "0.358752345"
    ],
    [
      "0.579548556",
      "0.075367370",
      "0.925653833",
      "0.423408042",
      "0.941589399",
      "0.578405675"
    ],
    [
      "0.170448672",
      "0.688457021",
      "0.782263226",
      "0.368615326",
      "0.706626401",
      "0.145975041"
    ]
  ]
}

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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