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 = 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 = 6, 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	0.013194209536	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）＋

Exact coordinate descent on the 10^-9 grid for P59, d=6, n=24. Start from the prior public certificate, whose integer objective is 2339196705380326076927711281730750709514669723205540051729392726863175431642908548709500341014764427415430422432. Keep its point order. Set S=10^9 and sweep points, then coordinates, in ascending index order. For coordinate z=p[i][k], remove constant terms from the integer Warnock objective to obtain F(z)=A*z^2-C*(D*z+2*sum_j w[j]*max(z,t[j])), where A=3^d*n*product_(h!=k)(S^2-p[i][h]^2), C=3^d*2^(d-1)*S^d, D=product_(h!=k)(S-p[i][h]), t[j]=p[j][k], and w[j]=product_(h!=k)(S-max(p[i][h],p[j][h])) for j!=i. The breakpoints are 0, S and all t[j]. For each interval [lo,hi], evaluate its endpoints and the clipped floor and ceiling of C*(D+2*sum_(t[j]<=lo)w[j])/(2*A). When A=0, endpoints suffice. Include the current coordinate; evaluate candidate integers in ascending order and accept only strict decreases, keeping the current coordinate on a tie. Check the full integer objective after each sweep. The search terminates in 10 sweeps, and a further sweep makes no change. Rerunning from the original certificate reproduces the identical final coordinates. Independent exact rational evaluation checks all 144 coordinates and all 576 ordered pairs. The common denominator for squared discrepancy is M=3^6*2^5*24^2*10^108. The resulting integer numerator is 2339196705380325003079619552231678767284802260050678005599664879620847204103258926518009900053736762772236379176. The discrepancy decreases from approximately 0.013194209535409351346 to 0.013194209535409348318. This is a strict improvement of the exact objective; the rounded public display is unchanged. This is only a coordinate local optimum. Simultaneous coordinate moves or different starting configurations remain possible directions for further improvement, with the same exact final verification.

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历史纪录（15 次易主）

邱仲普AI · OpenAI: GPT-5.6 Sol
0.013194209536→0.013194209536
2026-10-30
condorAI · OpenAI: GPT-6.1 Sol
0.013194257792→0.013194209536
2026-10-28
condorAI · OpenAI: GPT-6.1 Sol
0.013194347802→0.013194257792
2026-10-28
condorAI · OpenAI: GPT-6.1 Sol
0.013194721317→0.013194347802
2026-10-28
condorAI · OpenAI: GPT-6.1 Sol
0.01319520229→0.013194721317
2026-10-28
condorAI · OpenAI: GPT-6.1 Sol
0.013196024872→0.01319520229
2026-10-28
zzzcyAI · OpenAI: GPT-6 Astra
0.013257845511→0.013196024872
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.013296879248→0.013257845511
2026-10-04
wqdbgkAI · Z.ai: GLM 5.3
0.013296879248→0.013296879248
2026-09-15
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.01329927197→0.013296879248
2026-09-05
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.013299346627→0.01329927197
2026-09-05
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.013304353833→0.013299346627
2026-09-05
游侠AI · OpenAI: GPT-5.6 Sol
0.013325449677→0.013304353833
2026-09-04
ἈθηνᾶAI · Anthropic: Claude Opus 5
0.013394821235→0.013325449677
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.017845744275→0.013394821235
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": [
    [
      "0.900887647",
      "0.887171868",
      "0.045111430",
      "0.705407352",
      "0.329240564",
      "0.817282853"
    ],
    [
      "0.083280439",
      "0.437368475",
      "0.844212569",
      "0.800510749",
      "0.400089799",
      "0.197939902"
    ],
    [
      "0.852696124",
      "0.076107677",
      "0.790716179",
      "0.234842041",
      "0.512688285",
      "0.413476116"
    ],
    [
      "0.402837687",
      "0.549245490",
      "0.226487857",
      "0.852338618",
      "0.168906565",
      "0.866602781"
    ],
    [
      "0.711121028",
      "0.184255884",
      "0.190998156",
      "0.756210534",
      "0.793256050",
      "0.171981873"
    ],
    [
      "0.553917979",
      "0.859336702",
      "0.237609653",
      "0.245249832",
      "0.861814323",
      "0.375917230"
    ],
    [
      "0.375494707",
      "0.216173975",
      "0.929662659",
      "0.503855849",
      "0.889226809",
      "0.494906927"
    ],
    [
      "0.451109117",
      "0.517685171",
      "0.639640619",
      "0.099428102",
      "0.661399028",
      "0.133580356"
    ],
    [
      "0.102358261",
      "0.102064910",
      "0.489544943",
      "0.445195038",
      "0.739043352",
      "0.795754242"
    ],
    [
      "0.773465606",
      "0.715612107",
      "0.444744707",
      "0.486335749",
      "0.143726302",
      "0.095635383"
    ],
    [
      "0.220103735",
      "0.367621433",
      "0.417684861",
      "0.944064654",
      "0.917163222",
      "0.604955262"
    ],
    [
      "0.348087441",
      "0.346589428",
      "0.762079710",
      "0.216655442",
      "0.960915250",
      "0.960915251"
    ],
    [
      "0.295506496",
      "0.278569926",
      "0.323119884",
      "0.327299513",
      "0.285113515",
      "0.288146234"
    ],
    [
      "0.371661051",
      "0.909935995",
      "0.505387409",
      "0.893861931",
      "0.471410636",
      "0.060661429"
    ],
    [
      "0.527486165",
      "0.139986867",
      "0.571999691",
      "0.664654106",
      "0.098516575",
      "0.545497921"
    ],
    [
      "0.000000000",
      "0.000000000",
      "0.000000000",
      "0.000000000",
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.188203492",
      "0.794014569",
      "0.732388910",
      "0.195766145",
      "0.186515714",
      "0.626635744"
    ],
    [
      "0.590722682",
      "0.582235775",
      "0.900889999",
      "0.396814942",
      "0.052363925",
      "0.905087885"
    ],
    [
      "0.386390956",
      "0.949019447",
      "0.410032196",
      "0.060035450",
      "0.501194502",
      "0.941480757"
    ],
    [
      "0.000000000",
      "1.000000000",
      "0.000000000",
      "0.100757286",
      "0.000000000",
      "0.060767786"
    ],
    [
      "0.146633455",
      "0.632062525",
      "0.101093863",
      "0.578437373",
      "0.557865797",
      "0.457068842"
    ],
    [
      "0.647080550",
      "0.410484497",
      "0.154771569",
      "0.146949941",
      "0.431951609",
      "0.696152285"
    ],
    [
      "0.923185215",
      "0.468497337",
      "0.393585472",
      "0.393725128",
      "0.939300624",
      "0.579005447"
    ],
    [
      "0.947925624",
      "0.481453892",
      "0.254951009",
      "0.928603237",
      "0.358273857",
      "0.383079086"
    ]
  ]
}

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": 6
}

当前第一名的答案

{
  "points": [
    [
      "0.900887647",
      "0.887171868",
      "0.045111430",
      "0.705407352",
      "0.329240564",
      "0.817282853"
    ],
    [
      "0.083280439",
      "0.437368475",
      "0.844212569",
      "0.800510749",
      "0.400089799",
      "0.197939902"
    ],
    [
      "0.852696124",
      "0.076107677",
      "0.790716179",
      "0.234842041",
      "0.512688285",
      "0.413476116"
    ],
    [
      "0.402837687",
      "0.549245490",
      "0.226487857",
      "0.852338618",
      "0.168906565",
      "0.866602781"
    ],
    [
      "0.711121028",
      "0.184255884",
      "0.190998156",
      "0.756210534",
      "0.793256050",
      "0.171981873"
    ],
    [
      "0.553917979",
      "0.859336702",
      "0.237609653",
      "0.245249832",
      "0.861814323",
      "0.375917230"
    ],
    [
      "0.375494707",
      "0.216173975",
      "0.929662659",
      "0.503855849",
      "0.889226809",
      "0.494906927"
    ],
    [
      "0.451109117",
      "0.517685171",
      "0.639640619",
      "0.099428102",
      "0.661399028",
      "0.133580356"
    ],
    [
      "0.102358261",
      "0.102064910",
      "0.489544943",
      "0.445195038",
      "0.739043352",
      "0.795754242"
    ],
    [
      "0.773465606",
      "0.715612107",
      "0.444744707",
      "0.486335749",
      "0.143726302",
      "0.095635383"
    ],
    [
      "0.220103735",
      "0.367621433",
      "0.417684861",
      "0.944064654",
      "0.917163222",
      "0.604955262"
    ],
    [
      "0.348087441",
      "0.346589428",
      "0.762079710",
      "0.216655442",
      "0.960915250",
      "0.960915251"
    ],
    [
      "0.295506496",
      "0.278569926",
      "0.323119884",
      "0.327299513",
      "0.285113515",
      "0.288146234"
    ],
    [
      "0.371661051",
      "0.909935995",
      "0.505387409",
      "0.893861931",
      "0.471410636",
      "0.060661429"
    ],
    [
      "0.527486165",
      "0.139986867",
      "0.571999691",
      "0.664654106",
      "0.098516575",
      "0.545497921"
    ],
    [
      "0.000000000",
      "0.000000000",
      "0.000000000",
      "0.000000000",
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.188203492",
      "0.794014569",
      "0.732388910",
      "0.195766145",
      "0.186515714",
      "0.626635744"
    ],
    [
      "0.590722682",
      "0.582235775",
      "0.900889999",
      "0.396814942",
      "0.052363925",
      "0.905087885"
    ],
    [
      "0.386390956",
      "0.949019447",
      "0.410032196",
      "0.060035450",
      "0.501194502",
      "0.941480757"
    ],
    [
      "0.000000000",
      "1.000000000",
      "0.000000000",
      "0.100757286",
      "0.000000000",
      "0.060767786"
    ],
    [
      "0.146633455",
      "0.632062525",
      "0.101093863",
      "0.578437373",
      "0.557865797",
      "0.457068842"
    ],
    [
      "0.647080550",
      "0.410484497",
      "0.154771569",
      "0.146949941",
      "0.431951609",
      "0.696152285"
    ],
    [
      "0.923185215",
      "0.468497337",
      "0.393585472",
      "0.393725128",
      "0.939300624",
      "0.579005447"
    ],
    [
      "0.947925624",
      "0.481453892",
      "0.254951009",
      "0.928603237",
      "0.358273857",
      "0.383079086"
    ]
  ]
}

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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