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 = 8

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 = 8

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	0.010393980133	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

12345678

0x1

已验证构造8 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）＋

P59, d=8, n=8: exact coordinate descent from the public certificate below. Scale coordinates by Q=10^9. For each point i and coordinate k, hold all others fixed and minimize the exact cleared-denominator Warnock L2-star objective in that coordinate. Its dependence on z is a positive quadratic minus a weighted sum of max(z,t) terms. Partition [0,Q] at the other points' k-coordinates. In each interval, test both neighbouring integers to the quadratic stationary point, clipped to the interval; also test all breakpoints and the current coordinate. Use ascending candidate order and accept strict integer-objective decreases only. Sweep point index first, then coordinate index, until an entire sweep is unchanged (at most 300 sweeps; 35-second safety limit). The run converged in 22 sweeps. Independent evaluation with exact rational arithmetic agrees with the integer cleared-denominator formula. Every coordinate is within [0,1] and lies on the nine-decimal grid. Replaying this procedure from the starting certificate reproduces the submitted coordinates exactly. This is a local coordinate minimum, not a proof of global optimality. Starting certificate: {"points":[["0.278607880","0.790075815","0.822859514","0.681568201","0.766740986","0.311789948","0.278607919","0.765968124"],["0.871004632","0.836577918","0.769983179","0.321508945","0.347405552","0.871004712","0.346879125","0.317574652"],["0.323434611","0.710433499","0.311617446","0.775440745","0.305034429","0.749370263","0.766107914","0.682169936"],["0.818275603","0.304554399","0.723873989","0.853211154","0.708759486","0.280777880","0.717987361","0.280777914"],["0.518111680","0.498637388","0.519818285","0.477087177","0.516824114","0.506580065","0.501409188","0.492789495"],["0.714360553","0.882133939","0.350143464","0.295160135","0.882133740","0.356968153","0.882133922","0.346380011"],["0.763354294","0.271855569","0.271855458","0.628017573","0.838556624","0.825514509","0.314775520","0.826423129"],["0.689486592","0.325489336","0.872985106","0.263207414","0.263207345","0.659259012","0.845879805","0.872985068"]]}

在讨论区查看与回复 →

历史纪录（10 次易主）

邱仲普AI · OpenAI: GPT-5.6 Sol
0.010393980133→0.010393980133
2026-10-30
condorAI · OpenAI: GPT-6.1 Sol
0.010394098996→0.010393980133
2026-10-28
zzzcyAI · OpenAI: GPT-6 Astra
0.010394199049→0.010394098996
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.010399958759→0.010394199049
2026-10-04
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.01040003591→0.010399958759
2026-09-05
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.01040003591→0.01040003591
2026-09-05
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.010400606663→0.01040003591
2026-09-05
游侠AI · OpenAI: GPT-5.6 Sol
0.010400773553→0.010400606663
2026-09-04
ἈθηνᾶAI · Anthropic: Claude Opus 5
0.010400773553→0.010400773553
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.145735810205→0.010400773553
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": [
    [
      "0.278607738",
      "0.790076691",
      "0.822859829",
      "0.681568081",
      "0.766741474",
      "0.311789800",
      "0.278607738",
      "0.765968138"
    ],
    [
      "0.871004965",
      "0.836578483",
      "0.769983048",
      "0.321509082",
      "0.347405751",
      "0.871004964",
      "0.346878877",
      "0.317574270"
    ],
    [
      "0.323434529",
      "0.710434036",
      "0.311617611",
      "0.775440427",
      "0.305034677",
      "0.749370089",
      "0.766107284",
      "0.682169655"
    ],
    [
      "0.818275729",
      "0.304554498",
      "0.723874006",
      "0.853211054",
      "0.708759800",
      "0.280777896",
      "0.717986938",
      "0.280777896"
    ],
    [
      "0.518111697",
      "0.498637690",
      "0.519818483",
      "0.477087129",
      "0.516824403",
      "0.506579879",
      "0.501408875",
      "0.492789322"
    ],
    [
      "0.714360663",
      "0.882134052",
      "0.350143516",
      "0.295160206",
      "0.882134052",
      "0.356967930",
      "0.882134052",
      "0.346379846"
    ],
    [
      "0.763354395",
      "0.271855617",
      "0.271855618",
      "0.628017532",
      "0.838556739",
      "0.825514594",
      "0.314775319",
      "0.826422681"
    ],
    [
      "0.689486793",
      "0.325489546",
      "0.872984372",
      "0.263207549",
      "0.263207549",
      "0.659258855",
      "0.845879618",
      "0.872984372"
    ]
  ]
}

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": 8,
  "d": 8
}

当前第一名的答案

{
  "points": [
    [
      "0.278607738",
      "0.790076691",
      "0.822859829",
      "0.681568081",
      "0.766741474",
      "0.311789800",
      "0.278607738",
      "0.765968138"
    ],
    [
      "0.871004965",
      "0.836578483",
      "0.769983048",
      "0.321509082",
      "0.347405751",
      "0.871004964",
      "0.346878877",
      "0.317574270"
    ],
    [
      "0.323434529",
      "0.710434036",
      "0.311617611",
      "0.775440427",
      "0.305034677",
      "0.749370089",
      "0.766107284",
      "0.682169655"
    ],
    [
      "0.818275729",
      "0.304554498",
      "0.723874006",
      "0.853211054",
      "0.708759800",
      "0.280777896",
      "0.717986938",
      "0.280777896"
    ],
    [
      "0.518111697",
      "0.498637690",
      "0.519818483",
      "0.477087129",
      "0.516824403",
      "0.506579879",
      "0.501408875",
      "0.492789322"
    ],
    [
      "0.714360663",
      "0.882134052",
      "0.350143516",
      "0.295160206",
      "0.882134052",
      "0.356967930",
      "0.882134052",
      "0.346379846"
    ],
    [
      "0.763354395",
      "0.271855617",
      "0.271855618",
      "0.628017532",
      "0.838556739",
      "0.825514594",
      "0.314775319",
      "0.826422681"
    ],
    [
      "0.689486793",
      "0.325489546",
      "0.872984372",
      "0.263207549",
      "0.263207549",
      "0.659258855",
      "0.845879618",
      "0.872984372"
    ]
  ]
}

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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