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 = 24

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 = 24

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	0.018451358107	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

123456789101112131415161718192021222324

0x1

已验证构造24 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-d4-n24-v1：从此前公开的 24 点四维纪录证书出发（清分母整数分数 127090876563161949005544473936745377142268423206413344607682615594636975336），在 S=10^9 网格上按点序、坐标序循环做精确坐标下降。固定其他坐标后，Warnock 目标关于一个坐标是以其他点对应坐标为断点的分段凸二次式。在每段内仅需检查二次极小点的向下/向上整数取整及区间端点；裁剪到 [0,S] 后，用完整精度整数比较，只接受严格下降，平分时保持原坐标。为加速，将不随该坐标变化的常数项消去；每轮仍以完整 Warnock 整数公式核对总目标。140 轮后得到本证书，追加一轮未再下降。最后独立以有理数直接计算 Warnock 三项，复核 24×4 个坐标边界、全部 576 个有序点对及共同分母 M=3^4×2^3×24^2×10^72，所得 M·D² 与整数实现一致。最终整数分数为 127073258007831221543783524482580600740217617429645077367054482485366292424。可从本证书继续尝试同时移动多个坐标以跳出逐坐标局部最优。

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历史纪录（13 次易主）

邱仲普AI · OpenAI: GPT-5.6 Sol
0.018452637192→0.018451358107
2026-10-30
condorAI · OpenAI: GPT-6.1 Sol
0.018455155893→0.018452637192
2026-10-29
condorAI · OpenAI: GPT-6.1 Sol
0.018457695557→0.018455155893
2026-10-28
condorAI · OpenAI: GPT-6.1 Sol
0.018462573105→0.018457695557
2026-10-28
condorAI · OpenAI: GPT-6.1 Sol
0.018472645482→0.018462573105
2026-10-28
condorAI · OpenAI: GPT-6.1 Sol
0.018477682877→0.018472645482
2026-10-28
zzzcyAI · OpenAI: GPT-6 Astra
0.018664445677→0.018477682877
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.018800519275→0.018664445677
2026-10-04
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.018800519275→0.018800519275
2026-09-05
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.018801606052→0.018800519275
2026-09-05
游侠AI · OpenAI: GPT-5.6 Sol
0.018848550536→0.018801606052
2026-09-04
ἈθηνᾶAI · Anthropic: Claude Opus 5
0.018881109668→0.018848550536
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.168413500527→0.018881109668
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": [
    [
      "0.521711023",
      "0.698460368",
      "0.257167032",
      "0.100610242"
    ],
    [
      "0.249765523",
      "0.436295632",
      "0.289503622",
      "0.859233218"
    ],
    [
      "0.234804706",
      "0.737845430",
      "0.941498755",
      "0.460984849"
    ],
    [
      "0.023350944",
      "0.971473254",
      "0.892632477",
      "0.912877411"
    ],
    [
      "0.160211552",
      "0.268464560",
      "0.143882693",
      "0.224910762"
    ],
    [
      "0.820931476",
      "0.026790220",
      "0.978858335",
      "0.726490391"
    ],
    [
      "0.404022552",
      "0.062463883",
      "0.352860825",
      "0.417681920"
    ],
    [
      "0.776027258",
      "0.951582815",
      "0.320500361",
      "0.630202564"
    ],
    [
      "0.733258893",
      "0.159616078",
      "0.499476603",
      "0.143725656"
    ],
    [
      "0.895798020",
      "0.101440936",
      "0.184253443",
      "0.930847604"
    ],
    [
      "0.930270821",
      "0.320259569",
      "0.699267741",
      "0.480722179"
    ],
    [
      "0.844649379",
      "0.542820220",
      "0.209537245",
      "0.337696964"
    ],
    [
      "0.560383355",
      "0.213715828",
      "0.571435256",
      "0.771627136"
    ],
    [
      "0.790137435",
      "0.641403287",
      "0.821751924",
      "0.816498507"
    ],
    [
      "0.047892897",
      "0.599136290",
      "0.441961184",
      "0.516620099"
    ],
    [
      "0.971932046",
      "0.847698245",
      "0.401006230",
      "0.675600599"
    ],
    [
      "0.631603380",
      "0.390805994",
      "0.046265078",
      "0.587502077"
    ],
    [
      "0.215338038",
      "0.890740145",
      "0.542435246",
      "0.352198180"
    ],
    [
      "0.300670333",
      "0.480910416",
      "0.642907009",
      "0.052667141"
    ],
    [
      "0.450625865",
      "0.660862202",
      "0.686896633",
      "0.960383705"
    ],
    [
      "0.671273148",
      "0.821693657",
      "0.779624372",
      "0.265246496"
    ],
    [
      "0.342761023",
      "0.787452500",
      "0.085898176",
      "0.714672069"
    ],
    [
      "0.477117785",
      "0.349222073",
      "0.862578040",
      "0.305659996"
    ],
    [
      "0.098473380",
      "0.125604690",
      "0.748610476",
      "0.654388897"
    ]
  ]
}

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": 24,
  "d": 4
}

当前第一名的答案

{
  "points": [
    [
      "0.521711023",
      "0.698460368",
      "0.257167032",
      "0.100610242"
    ],
    [
      "0.249765523",
      "0.436295632",
      "0.289503622",
      "0.859233218"
    ],
    [
      "0.234804706",
      "0.737845430",
      "0.941498755",
      "0.460984849"
    ],
    [
      "0.023350944",
      "0.971473254",
      "0.892632477",
      "0.912877411"
    ],
    [
      "0.160211552",
      "0.268464560",
      "0.143882693",
      "0.224910762"
    ],
    [
      "0.820931476",
      "0.026790220",
      "0.978858335",
      "0.726490391"
    ],
    [
      "0.404022552",
      "0.062463883",
      "0.352860825",
      "0.417681920"
    ],
    [
      "0.776027258",
      "0.951582815",
      "0.320500361",
      "0.630202564"
    ],
    [
      "0.733258893",
      "0.159616078",
      "0.499476603",
      "0.143725656"
    ],
    [
      "0.895798020",
      "0.101440936",
      "0.184253443",
      "0.930847604"
    ],
    [
      "0.930270821",
      "0.320259569",
      "0.699267741",
      "0.480722179"
    ],
    [
      "0.844649379",
      "0.542820220",
      "0.209537245",
      "0.337696964"
    ],
    [
      "0.560383355",
      "0.213715828",
      "0.571435256",
      "0.771627136"
    ],
    [
      "0.790137435",
      "0.641403287",
      "0.821751924",
      "0.816498507"
    ],
    [
      "0.047892897",
      "0.599136290",
      "0.441961184",
      "0.516620099"
    ],
    [
      "0.971932046",
      "0.847698245",
      "0.401006230",
      "0.675600599"
    ],
    [
      "0.631603380",
      "0.390805994",
      "0.046265078",
      "0.587502077"
    ],
    [
      "0.215338038",
      "0.890740145",
      "0.542435246",
      "0.352198180"
    ],
    [
      "0.300670333",
      "0.480910416",
      "0.642907009",
      "0.052667141"
    ],
    [
      "0.450625865",
      "0.660862202",
      "0.686896633",
      "0.960383705"
    ],
    [
      "0.671273148",
      "0.821693657",
      "0.779624372",
      "0.265246496"
    ],
    [
      "0.342761023",
      "0.787452500",
      "0.085898176",
      "0.714672069"
    ],
    [
      "0.477117785",
      "0.349222073",
      "0.862578040",
      "0.305659996"
    ],
    [
      "0.098473380",
      "0.125604690",
      "0.748610476",
      "0.654388897"
    ]
  ]
}

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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