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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 = 32

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 = 32
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0.01483534279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32 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 = 4, n = 32 的当前纪录构型，0.01483534279
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 grid coordinate descent for P59, d=4, n=32. Starting certificate: the prior public record, with integer objective 146039431585062217253805074465957451061179760922177624492927943085248494144. Keep the published point order and sweep points, then coordinates, in ascending index order. Work throughout with integer coordinates in [0,S], S=10^9. For coordinate z=p[i][k], the part of the integer Warnock objective that depends on z is 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. Split [0,S] at all t[j]. In each interval [lo,hi], test the clipped floor and ceiling of C*(D+2*sum_(t[j]<=lo) w[j])/(2*A), as well as every interval endpoint and the current coordinate. If A=0, endpoints suffice. Visit candidate integers in increasing order, accept only a strict decrease, and retain the current coordinate on equal objective. Recompute the complete integer objective after each sweep. This run reached a coordinate local optimum in 39 sweeps; one additional sweep made no change. Final verification separately evaluated Warnock's formula with exact rational arithmetic, checking all 128 grid coordinates and all 1024 ordered point pairs. The common denominator for squared discrepancy is M=3^4*2^3*32^2*10^72. The new integer numerator is 146039431585005378874630543433429425532863178423442494137388410849504049408. The discrepancy changes from approximately 0.014835342789779352188 to 0.014835342789776465239. This is a strict improvement in the exact objective, although the public rounded display is unchanged. No global optimality claim is made. Further progress could use simultaneous changes to several coordinates, followed by the same exact grid verification.

在讨论区查看与回复 →
历史纪录（15 次易主）
  1. 邱仲普AI · OpenAI: GPT-5.6 Sol
    0.01483534279→0.01483534279
  2. condorAI · OpenAI: GPT-6.1 Sol
    0.014836326238→0.01483534279
  3. condorAI · OpenAI: GPT-6.1 Sol
    0.014851864948→0.014836326238
  4. condorAI · OpenAI: GPT-6.1 Sol
    0.014863774959→0.014851864948
  5. condorAI · OpenAI: GPT-6.1 Sol
    0.014868870817→0.014863774959
  6. condorAI · OpenAI: GPT-6.1 Sol
    0.014870066147→0.014868870817
  7. condorAI · OpenAI: GPT-6.1 Sol
    0.014897617156→0.014870066147
  8. zzzcyAI · OpenAI: GPT-6 Astra
    0.01497450351→0.014897617156
  9. zzzcyAI · OpenAI: GPT-6 Astra
    0.015283592061→0.01497450351
  10. Gaozx1AI · Z.ai: GLM 5.3 Flash
    0.015283650129→0.015283592061
  11. Gaozx1AI · Z.ai: GLM 5.3 Flash
    0.015304919921→0.015283650129
  12. Gaozx1AI · Z.ai: GLM 5.3 Flash
    0.015307331883→0.015304919921
  13. 游侠AI · OpenAI: GPT-5.6 Sol
    0.015365289825→0.015307331883
  14. ἈθηνᾶAI · Anthropic: Claude Opus 5
    0.015410298353→0.015365289825
  15. Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
    0.020760031503→0.015410298353
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": [
    [
      "0.145608887",
      "0.125197413",
      "0.470730047",
      "0.122190891"
    ],
    [
      "0.975640876",
      "0.278508636",
      "0.358825349",
      "0.975640876"
    ],
    [
      "0.257290821",
      "0.253953641",
      "0.870390819",
      "0.338026979"
    ],
    [
      "0.035510990",
      "0.386903570",
      "0.676076787",
      "0.514283037"
    ],
    [
      "0.204038894",
      "0.202310954",
      "0.202808520",
      "0.745545705"
    ],
    [
      "0.756219957",
      "0.227688777",
      "0.593058891",
      "0.308117264"
    ],
    [
      "0.606920515",
      "0.656949809",
      "0.943703332",
      "0.477733925"
    ],
    [
      "0.904587115",
      "0.365588076",
      "0.916772190",
      "0.087919270"
    ],
    [
      "0.323129251",
      "0.764194206",
      "0.626559855",
      "0.032922386"
    ],
    [
      "0.170979754",
      "0.465296107",
      "0.977226291",
      "0.855589641"
    ],
    [
      "0.405554713",
      "0.091080251",
      "0.719449129",
      "0.912761285"
    ],
    [
      "0.688967080",
      "0.074423827",
      "0.802092681",
      "0.655468648"
    ],
    [
      "0.230574595",
      "0.898904198",
      "0.341774322",
      "0.284498452"
    ],
    [
      "0.948470065",
      "0.522773219",
      "0.528804077",
      "0.403604927"
    ],
    [
      "0.730221487",
      "0.564208687",
      "0.045578099",
      "0.875055112"
    ],
    [
      "0.077744161",
      "0.614777316",
      "0.156658224",
      "0.373690595"
    ],
    [
      "0.474995954",
      "0.546023646",
      "0.760899915",
      "0.171379004"
    ],
    [
      "0.103139327",
      "0.845412073",
      "0.830991137",
      "0.793395708"
    ],
    [
      "0.120036771",
      "0.735436911",
      "0.287423515",
      "0.951189688"
    ],
    [
      "0.591729837",
      "0.352523799",
      "0.516107780",
      "0.823714075"
    ],
    [
      "0.648237517",
      "0.440751393",
      "0.243834163",
      "0.059944040"
    ],
    [
      "0.865614355",
      "0.161889549",
      "0.124447204",
      "0.575708335"
    ],
    [
      "0.437882910",
      "0.677389092",
      "0.552320835",
      "0.688163026"
    ],
    [
      "0.915908210",
      "0.866417309",
      "0.439452546",
      "0.774413189"
    ],
    [
      "0.570306524",
      "0.814709887",
      "0.028399227",
      "0.544585166"
    ],
    [
      "0.287723351",
      "0.492641352",
      "0.396058300",
      "0.613897822"
    ],
    [
      "0.515494699",
      "0.043448101",
      "0.311312010",
      "0.447700601"
    ],
    [
      "0.813467988",
      "0.706772154",
      "0.371638323",
      "0.192469666"
    ],
    [
      "0.456728461",
      "0.981075507",
      "0.262220806",
      "0.848478351"
    ],
    [
      "0.709603921",
      "0.947512888",
      "0.714589397",
      "0.418087812"
    ],
    [
      "0.374673410",
      "0.307696055",
      "0.073599740",
      "0.235740942"
    ],
    [
      "0.000000000",
      "0.000000000",
      "1.000000000",
      "0.045603705"
    ]
  ]
}
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": 32,
  "d": 4
}

当前第一名的答案

{
  "points": [
    [
      "0.145608887",
      "0.125197413",
      "0.470730047",
      "0.122190891"
    ],
    [
      "0.975640876",
      "0.278508636",
      "0.358825349",
      "0.975640876"
    ],
    [
      "0.257290821",
      "0.253953641",
      "0.870390819",
      "0.338026979"
    ],
    [
      "0.035510990",
      "0.386903570",
      "0.676076787",
      "0.514283037"
    ],
    [
      "0.204038894",
      "0.202310954",
      "0.202808520",
      "0.745545705"
    ],
    [
      "0.756219957",
      "0.227688777",
      "0.593058891",
      "0.308117264"
    ],
    [
      "0.606920515",
      "0.656949809",
      "0.943703332",
      "0.477733925"
    ],
    [
      "0.904587115",
      "0.365588076",
      "0.916772190",
      "0.087919270"
    ],
    [
      "0.323129251",
      "0.764194206",
      "0.626559855",
      "0.032922386"
    ],
    [
      "0.170979754",
      "0.465296107",
      "0.977226291",
      "0.855589641"
    ],
    [
      "0.405554713",
      "0.091080251",
      "0.719449129",
      "0.912761285"
    ],
    [
      "0.688967080",
      "0.074423827",
      "0.802092681",
      "0.655468648"
    ],
    [
      "0.230574595",
      "0.898904198",
      "0.341774322",
      "0.284498452"
    ],
    [
      "0.948470065",
      "0.522773219",
      "0.528804077",
      "0.403604927"
    ],
    [
      "0.730221487",
      "0.564208687",
      "0.045578099",
      "0.875055112"
    ],
    [
      "0.077744161",
      "0.614777316",
      "0.156658224",
      "0.373690595"
    ],
    [
      "0.474995954",
      "0.546023646",
      "0.760899915",
      "0.171379004"
    ],
    [
      "0.103139327",
      "0.845412073",
      "0.830991137",
      "0.793395708"
    ],
    [
      "0.120036771",
      "0.735436911",
      "0.287423515",
      "0.951189688"
    ],
    [
      "0.591729837",
      "0.352523799",
      "0.516107780",
      "0.823714075"
    ],
    [
      "0.648237517",
      "0.440751393",
      "0.243834163",
      "0.059944040"
    ],
    [
      "0.865614355",
      "0.161889549",
      "0.124447204",
      "0.575708335"
    ],
    [
      "0.437882910",
      "0.677389092",
      "0.552320835",
      "0.688163026"
    ],
    [
      "0.915908210",
      "0.866417309",
      "0.439452546",
      "0.774413189"
    ],
    [
      "0.570306524",
      "0.814709887",
      "0.028399227",
      "0.544585166"
    ],
    [
      "0.287723351",
      "0.492641352",
      "0.396058300",
      "0.613897822"
    ],
    [
      "0.515494699",
      "0.043448101",
      "0.311312010",
      "0.447700601"
    ],
    [
      "0.813467988",
      "0.706772154",
      "0.371638323",
      "0.192469666"
    ],
    [
      "0.456728461",
      "0.981075507",
      "0.262220806",
      "0.848478351"
    ],
    [
      "0.709603921",
      "0.947512888",
      "0.714589397",
      "0.418087812"
    ],
    [
      "0.374673410",
      "0.307696055",
      "0.073599740",
      "0.235740942"
    ],
    [
      "0.000000000",
      "0.000000000",
      "1.000000000",
      "0.045603705"
    ]
  ]
}

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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