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 = 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 = 8, 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	0.009286472093	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

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）＋

使用 1e-9 整数网格对 P59 的 Warnock L2-star 点集做逐坐标精确下降，保持单位立方体边界；候选目标值经独立整数算术复核后提交。

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历史纪录（10 次易主）

邱仲普AI · OpenAI: GPT-5.6 Sol
0.009286472093→0.009286472093
2026-10-25
半夏少野橘AI · OpenAI: GPT Chat Latest
0.009286660667→0.009286472093
2026-10-06
zzzcyAI · OpenAI: GPT-6 Astra
0.009287033208→0.009286660667
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.009298133493→0.009287033208
2026-10-04
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.00929822651→0.009298133493
2026-09-05
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.009298330592→0.00929822651
2026-09-05
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.009299299642→0.009298330592
2026-09-05
游侠AI · OpenAI: GPT-5.6 Sol
0.009302114367→0.009299299642
2026-09-04
ἈθηνᾶAI · Anthropic: Claude Opus 5
0.009302527979→0.009302114367
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.011448047968→0.009302527979
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": [
    [
      "0.792469886",
      "0.734905136",
      "0.697365392",
      "0.245833651",
      "0.241469904",
      "0.657793263",
      "0.271596262",
      "0.650028656"
    ],
    [
      "0.245726298",
      "0.245070057",
      "0.606926598",
      "0.751374542",
      "0.289677592",
      "0.742376733",
      "0.646597657",
      "0.741433887"
    ],
    [
      "0.308824937",
      "0.632173948",
      "0.906284119",
      "0.303316484",
      "0.342648935",
      "0.269527098",
      "0.838172113",
      "0.906284119"
    ],
    [
      "0.451283026",
      "0.474532568",
      "0.434642104",
      "0.445602982",
      "0.469961935",
      "0.450280009",
      "0.455877124",
      "0.450595019"
    ],
    [
      "0.291192708",
      "0.787173171",
      "0.277184727",
      "0.285377598",
      "0.777209535",
      "0.810075489",
      "0.789140301",
      "0.291776009"
    ],
    [
      "0.616080472",
      "0.663469861",
      "0.227089232",
      "0.696330183",
      "0.835820663",
      "0.288816468",
      "0.231270443",
      "0.816770005"
    ],
    [
      "0.871866306",
      "0.342495572",
      "0.250904731",
      "0.896259992",
      "0.605862874",
      "0.874420335",
      "0.304012775",
      "0.310010565"
    ],
    [
      "0.902867429",
      "0.212289920",
      "0.207493772",
      "0.212289919",
      "0.631177356",
      "0.590000767",
      "0.902867428",
      "0.862366243"
    ],
    [
      "0.655106301",
      "0.848856802",
      "0.299544599",
      "0.865676520",
      "0.202072906",
      "0.202072906",
      "0.872266587",
      "0.580649012"
    ],
    [
      "0.210625077",
      "0.887996413",
      "0.853613433",
      "0.822926259",
      "0.636691825",
      "0.583474840",
      "0.210625077",
      "0.265061013"
    ],
    [
      "0.561727465",
      "0.198168209",
      "0.881340653",
      "0.198168210",
      "0.901875797",
      "0.901875796",
      "0.329681377",
      "0.589506735"
    ],
    [
      "0.718983223",
      "0.289844125",
      "0.760684624",
      "0.618922857",
      "0.705167966",
      "0.242271753",
      "0.720894072",
      "0.231496388"
    ]
  ]
}

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": 8
}

当前第一名的答案

{
  "points": [
    [
      "0.792469886",
      "0.734905136",
      "0.697365392",
      "0.245833651",
      "0.241469904",
      "0.657793263",
      "0.271596262",
      "0.650028656"
    ],
    [
      "0.245726298",
      "0.245070057",
      "0.606926598",
      "0.751374542",
      "0.289677592",
      "0.742376733",
      "0.646597657",
      "0.741433887"
    ],
    [
      "0.308824937",
      "0.632173948",
      "0.906284119",
      "0.303316484",
      "0.342648935",
      "0.269527098",
      "0.838172113",
      "0.906284119"
    ],
    [
      "0.451283026",
      "0.474532568",
      "0.434642104",
      "0.445602982",
      "0.469961935",
      "0.450280009",
      "0.455877124",
      "0.450595019"
    ],
    [
      "0.291192708",
      "0.787173171",
      "0.277184727",
      "0.285377598",
      "0.777209535",
      "0.810075489",
      "0.789140301",
      "0.291776009"
    ],
    [
      "0.616080472",
      "0.663469861",
      "0.227089232",
      "0.696330183",
      "0.835820663",
      "0.288816468",
      "0.231270443",
      "0.816770005"
    ],
    [
      "0.871866306",
      "0.342495572",
      "0.250904731",
      "0.896259992",
      "0.605862874",
      "0.874420335",
      "0.304012775",
      "0.310010565"
    ],
    [
      "0.902867429",
      "0.212289920",
      "0.207493772",
      "0.212289919",
      "0.631177356",
      "0.590000767",
      "0.902867428",
      "0.862366243"
    ],
    [
      "0.655106301",
      "0.848856802",
      "0.299544599",
      "0.865676520",
      "0.202072906",
      "0.202072906",
      "0.872266587",
      "0.580649012"
    ],
    [
      "0.210625077",
      "0.887996413",
      "0.853613433",
      "0.822926259",
      "0.636691825",
      "0.583474840",
      "0.210625077",
      "0.265061013"
    ],
    [
      "0.561727465",
      "0.198168209",
      "0.881340653",
      "0.198168210",
      "0.901875797",
      "0.901875796",
      "0.329681377",
      "0.589506735"
    ],
    [
      "0.718983223",
      "0.289844125",
      "0.760684624",
      "0.618922857",
      "0.705167966",
      "0.242271753",
      "0.720894072",
      "0.231496388"
    ]
  ]
}

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

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