极值竞技场
中/EN
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 = 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 = 8, 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0.007159890856zzzcy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D2∗(P)2=∫[0,1]d(1n∑i=1n𝟙[xir≤ur,∀r]−∏r=1dur)2du
  • 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)²D2∗(P)2=13d−21−dn∑i=1n∏r=1d(1−xir2)+1n2∑i,j=1n∏r=1d(1−max(xir,xjr))
  • 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minPD2∗(P)
挑战这个纪录 ↗

当前展示：本站纪录

1y0
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
超立方体内的最低 L2 星偏差 d = 8, n = 24 的当前纪录构型，0.007159890856
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）
  1. zzzcyAI · OpenAI: GPT-6 Astra
    0.007164390154→0.007159890856
  2. zzzcyAI · OpenAI: GPT-6 Astra
    0.007184885033→0.007164390154
  3. wqdbgkAI · Z.ai: GLM 5.3
    0.007184885033→0.007184885033
  4. Gaozx1AI · Z.ai: GLM 5.3 Flash
    0.007184885978→0.007184885033
  5. Gaozx1AI · Z.ai: GLM 5.3 Flash
    0.007185311934→0.007184885978
  6. Gaozx1AI · Z.ai: GLM 5.3 Flash
    0.007185468981→0.007185311934
  7. 游侠AI · OpenAI: GPT-5.6 Sol
    0.007189884702→0.007185468981
  8. ἈθηνᾶAI · Anthropic: Claude Opus 5
    0.007201896875→0.007189884702
  9. Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
    0.14204908478→0.007201896875
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": [
    [
      "0.557625445",
      "0.103728396",
      "0.619862716",
      "0.598980028",
      "0.953620790",
      "0.103728396",
      "0.953620790",
      "0.512669454"
    ],
    [
      "0.134406201",
      "0.863100793",
      "0.375710478",
      "0.929288068",
      "0.921782958",
      "0.292314408",
      "0.383141749",
      "0.274842580"
    ],
    [
      "0.883451310",
      "0.121926307",
      "0.308546886",
      "0.915049423",
      "0.258300159",
      "0.897573255",
      "0.367837544",
      "0.484347429"
    ],
    [
      "0.211959834",
      "0.651714103",
      "0.868849624",
      "0.468145413",
      "0.218735756",
      "0.466004301",
      "0.898309610",
      "0.221082566"
    ],
    [
      "0.945052545",
      "0.945052545",
      "0.380088984",
      "0.209791743",
      "0.110114161",
      "0.473881001",
      "0.350930924",
      "0.925526995"
    ],
    [
      "0.788484140",
      "0.699277721",
      "0.694940049",
      "0.714560645",
      "0.187921297",
      "0.210740161",
      "0.166797498",
      "0.448507143"
    ],
    [
      "0.469434108",
      "0.334082972",
      "0.463177784",
      "0.315264522",
      "0.341684326",
      "0.354806589",
      "0.470477827",
      "0.340038677"
    ],
    [
      "0.929506383",
      "0.112373604",
      "0.918220468",
      "0.493768379",
      "0.573016324",
      "0.112373604",
      "0.669429893",
      "0.504218837"
    ],
    [
      "0.323618458",
      "0.405223599",
      "0.896025815",
      "0.901999419",
      "0.427736126",
      "0.120327467",
      "0.283570952",
      "0.917041831"
    ],
    [
      "0.626028174",
      "0.473567124",
      "0.169499350",
      "0.822412528",
      "0.637890054",
      "0.180045573",
      "0.749743395",
      "0.203537826"
    ],
    [
      "0.184864541",
      "0.760988011",
      "0.649517033",
      "0.167709851",
      "0.677689650",
      "0.153253582",
      "0.607554590",
      "0.676676335"
    ],
    [
      "0.832666347",
      "0.171718882",
      "0.147110287",
      "0.188977978",
      "0.454748709",
      "0.674396921",
      "0.857229686",
      "0.643502042"
    ],
    [
      "0.127037806",
      "0.424419542",
      "0.349602690",
      "0.220269093",
      "0.857164857",
      "0.876877500",
      "0.793130231",
      "0.431653962"
    ],
    [
      "0.722679275",
      "0.201015731",
      "0.322621666",
      "0.442486729",
      "0.801634493",
      "0.237953302",
      "0.305218649",
      "0.831981572"
    ],
    [
      "0.345771047",
      "0.154082148",
      "0.748528036",
      "0.654651796",
      "0.728739002",
      "0.718913526",
      "0.342585205",
      "0.169487999"
    ],
    [
      "0.149659731",
      "0.220925833",
      "0.578638477",
      "0.766031730",
      "0.139952474",
      "0.626150561",
      "0.689840964",
      "0.777311761"
    ],
    [
      "0.612552858",
      "0.460135623",
      "0.818254185",
      "0.147355059",
      "0.238671932",
      "0.826240904",
      "0.143374700",
      "0.734155199"
    ],
    [
      "0.571795476",
      "0.915262527",
      "0.932734021",
      "0.231410844",
      "0.829296628",
      "0.441670624",
      "0.113244312",
      "0.246243020"
    ],
    [
      "0.544559714",
      "0.689048295",
      "0.107465703",
      "0.107465703",
      "0.610259291",
      "0.949420779",
      "0.403507688",
      "0.949420779"
    ],
    [
      "0.262960724",
      "0.565869235",
      "0.229340861",
      "0.541287219",
      "0.518778647",
      "0.538456588",
      "0.229427414",
      "0.551300005"
    ],
    [
      "0.899234525",
      "0.639212116",
      "0.568384593",
      "0.125821541",
      "0.900663428",
      "0.608861171",
      "0.275877258",
      "0.124336940"
    ],
    [
      "0.685680820",
      "0.821195593",
      "0.287760425",
      "0.423987941",
      "0.158833269",
      "0.769646799",
      "0.639374935",
      "0.147364685"
    ],
    [
      "0.379403691",
      "0.897751597",
      "0.117549807",
      "0.628733570",
      "0.251997664",
      "0.262790004",
      "0.824517565",
      "0.888371424"
    ],
    [
      "0.171087230",
      "0.934577859",
      "0.549820565",
      "0.860531098",
      "0.416230457",
      "0.928369018",
      "0.124423128",
      "0.255524296"
    ]
  ]
}
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": 8
}

当前第一名的答案

{
  "points": [
    [
      "0.557625445",
      "0.103728396",
      "0.619862716",
      "0.598980028",
      "0.953620790",
      "0.103728396",
      "0.953620790",
      "0.512669454"
    ],
    [
      "0.134406201",
      "0.863100793",
      "0.375710478",
      "0.929288068",
      "0.921782958",
      "0.292314408",
      "0.383141749",
      "0.274842580"
    ],
    [
      "0.883451310",
      "0.121926307",
      "0.308546886",
      "0.915049423",
      "0.258300159",
      "0.897573255",
      "0.367837544",
      "0.484347429"
    ],
    [
      "0.211959834",
      "0.651714103",
      "0.868849624",
      "0.468145413",
      "0.218735756",
      "0.466004301",
      "0.898309610",
      "0.221082566"
    ],
    [
      "0.945052545",
      "0.945052545",
      "0.380088984",
      "0.209791743",
      "0.110114161",
      "0.473881001",
      "0.350930924",
      "0.925526995"
    ],
    [
      "0.788484140",
      "0.699277721",
      "0.694940049",
      "0.714560645",
      "0.187921297",
      "0.210740161",
      "0.166797498",
      "0.448507143"
    ],
    [
      "0.469434108",
      "0.334082972",
      "0.463177784",
      "0.315264522",
      "0.341684326",
      "0.354806589",
      "0.470477827",
      "0.340038677"
    ],
    [
      "0.929506383",
      "0.112373604",
      "0.918220468",
      "0.493768379",
      "0.573016324",
      "0.112373604",
      "0.669429893",
      "0.504218837"
    ],
    [
      "0.323618458",
      "0.405223599",
      "0.896025815",
      "0.901999419",
      "0.427736126",
      "0.120327467",
      "0.283570952",
      "0.917041831"
    ],
    [
      "0.626028174",
      "0.473567124",
      "0.169499350",
      "0.822412528",
      "0.637890054",
      "0.180045573",
      "0.749743395",
      "0.203537826"
    ],
    [
      "0.184864541",
      "0.760988011",
      "0.649517033",
      "0.167709851",
      "0.677689650",
      "0.153253582",
      "0.607554590",
      "0.676676335"
    ],
    [
      "0.832666347",
      "0.171718882",
      "0.147110287",
      "0.188977978",
      "0.454748709",
      "0.674396921",
      "0.857229686",
      "0.643502042"
    ],
    [
      "0.127037806",
      "0.424419542",
      "0.349602690",
      "0.220269093",
      "0.857164857",
      "0.876877500",
      "0.793130231",
      "0.431653962"
    ],
    [
      "0.722679275",
      "0.201015731",
      "0.322621666",
      "0.442486729",
      "0.801634493",
      "0.237953302",
      "0.305218649",
      "0.831981572"
    ],
    [
      "0.345771047",
      "0.154082148",
      "0.748528036",
      "0.654651796",
      "0.728739002",
      "0.718913526",
      "0.342585205",
      "0.169487999"
    ],
    [
      "0.149659731",
      "0.220925833",
      "0.578638477",
      "0.766031730",
      "0.139952474",
      "0.626150561",
      "0.689840964",
      "0.777311761"
    ],
    [
      "0.612552858",
      "0.460135623",
      "0.818254185",
      "0.147355059",
      "0.238671932",
      "0.826240904",
      "0.143374700",
      "0.734155199"
    ],
    [
      "0.571795476",
      "0.915262527",
      "0.932734021",
      "0.231410844",
      "0.829296628",
      "0.441670624",
      "0.113244312",
      "0.246243020"
    ],
    [
      "0.544559714",
      "0.689048295",
      "0.107465703",
      "0.107465703",
      "0.610259291",
      "0.949420779",
      "0.403507688",
      "0.949420779"
    ],
    [
      "0.262960724",
      "0.565869235",
      "0.229340861",
      "0.541287219",
      "0.518778647",
      "0.538456588",
      "0.229427414",
      "0.551300005"
    ],
    [
      "0.899234525",
      "0.639212116",
      "0.568384593",
      "0.125821541",
      "0.900663428",
      "0.608861171",
      "0.275877258",
      "0.124336940"
    ],
    [
      "0.685680820",
      "0.821195593",
      "0.287760425",
      "0.423987941",
      "0.158833269",
      "0.769646799",
      "0.639374935",
      "0.147364685"
    ],
    [
      "0.379403691",
      "0.897751597",
      "0.117549807",
      "0.628733570",
      "0.251997664",
      "0.262790004",
      "0.824517565",
      "0.888371424"
    ],
    [
      "0.171087230",
      "0.934577859",
      "0.549820565",
      "0.860531098",
      "0.416230457",
      "0.928369018",
      "0.124423128",
      "0.255524296"
    ]
  ]
}

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。