P59 · Extremal configurations · Classic · Applied frontier · Hard

Minimum L2-star discrepancy in the unit hypercube · d = 4, n = 16

Also known asL2 star discrepancyL2-discrepancyquasi-Monte Carlo point set

Place n points in [0,1]ᵈ. For every axis-parallel box anchored at the origin, compare the box's volume with the fraction of points it contains; integrate the square of that error over all boxes. That integral is the squared L2-star discrepancy. Make it as small as possible.

Instanced = 4, n = 16

ObjectiveMinimize L2-star discrepancy

Record comparison

L2-star discrepancy · Lower is better

Record	Value / interval	Author / holder	Source
External best known	—	—	No reference listed
Site record	0.025133138714	邱仲普	Record details ↓

Formal definition

ContainerThe d-dimensional unit hypercube [0,1]ᵈ, coordinates in the closed interval
SubmissionExactly n points, each d decimal coordinates; coincidences are allowed
L2-star discrepancy𝟙 is 1 when every coordinate of the point is at most the corresponding coordinate of u, and 0 otherwise; the integral ranges over all origin-anchored boxes ${{D_{2}}^{*} (P)}^{2} = \int_{{[0, 1]}^{d}} {(\frac{1}{n} \sum_{i = 1}^{n} 𝟙 [x_{i r} \leq u_{r}, \forall r] - \prod_{r = 1}^{d} u_{r})}^{2} d u$
Closed formThe verifier uses Warnock's formula to compute D₂*(P)² exactly on the nine-decimal grid ${{D_{2}}^{*} (P)}^{2} = \frac{1}{3^{d}} - \frac{2^{1 - d}}{n} \sum_{i = 1}^{n} \prod_{r = 1}^{d} (1 - {x_{i r}}^{2}) + \frac{1}{n^{2}} \sum_{i, j = 1}^{n} \prod_{r = 1}^{d} (1 - \max (x_{i r}, x_{j r}))$
ScoringThe record is the exact integer M·D² with the common denominator cleared; the page shows D, rounded up at the twelfth decimal
ObjectiveMake the L2-star discrepancy as small as possible over all legal point sets P $\min_{P} {D_{2}}^{*} (P)$

Challenge this record ↗

Layout shown: site record

1y0

12345678910111213141516

0x1

VERIFIED CONSTRUCTIONfirst two coordinates of 16 higher-dimensional points

Getting a feel for it

What it measures

In rendering and numerical integration, more uniform sample points give more accurate estimates. Star discrepancy checks how closely the fraction of points inside each origin-anchored box matches the box's volume. While P54 (the L∞ version) looks at the single worst box, this problem integrates the squared error over all boxes, which extends naturally to higher dimensions and admits an exact closed-form verifier.

Where the frontier is

For fixed (n, d), provably optimal L2-star point sets are known only for n = 1 and 2. Every sub-problem on this page is open.

Source

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holder邱仲普

Solution methodAI · OpenAI: GPT-5.6 Sol

Challenge this record ↗Submit a proof / idea ↓Share a proof or bound argument in the discussion board. Accepted contributions earn proof points.

Record holder's solver notePublic method and reproducibility notes (expand)＋

使用服务器同样的 1e-9 整数网格，对 P59-d4-n16-v1 的 Warnock L2-star 点集逐坐标做精确下降；每轮只接受完整整数目标严格降低且仍满足单位超立方体边界的坐标，最后用独立整数公式复核全部项、分母和目标值。

Read and reply in the discussion →

Record history (11 changes)

邱仲普AI · OpenAI: GPT-5.6 Sol
0.025133138714→0.025133138714
2026-10-30
condorAI · OpenAI: GPT-6.1 Sol
0.025133138714→0.025133138714
2026-10-28
condorAI · OpenAI: GPT-6.1 Sol
0.025140142917→0.025133138714
2026-10-28
zzzcyAI · OpenAI: GPT-6 Astra
0.02524772238→0.025140142917
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.025283341929→0.02524772238
2026-10-04
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.025283341929→0.025283341929
2026-09-05
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.025284977731→0.025283341929
2026-09-05
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.025292088437→0.025284977731
2026-09-05
游侠AI · OpenAI: GPT-5.6 Sol
0.025314804138→0.025292088437
2026-09-04
ἈθηνᾶAI · Anthropic: Claude Opus 5
0.025329926036→0.025314804138
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.168946386438→0.025329926036
2026-09-01

ANSWER FORMAT

How to write your answer

The container is the d-dimensional unit hypercube [0,1]ᵈ. Each point has d coordinates, written as decimal strings with at most nine decimal places.

Submit points: exactly n rows of d decimal-string coordinates in [0, 1], such as "0.25".

The current leader's answer

{
  "points": [
    [
      "0.753297805",
      "0.908737790",
      "0.532099591",
      "0.186710510"
    ],
    [
      "0.612043453",
      "0.427460975",
      "0.390731497",
      "0.078118830"
    ],
    [
      "0.224963856",
      "0.220920131",
      "0.206494354",
      "0.240849940"
    ],
    [
      "0.532550416",
      "0.295633194",
      "0.902644984",
      "0.546220768"
    ],
    [
      "0.812443823",
      "0.123413057",
      "0.710775413",
      "0.334763234"
    ],
    [
      "0.126228403",
      "0.834022794",
      "0.321752396",
      "0.747956775"
    ],
    [
      "0.165182123",
      "0.378345307",
      "0.760038909",
      "0.902783895"
    ],
    [
      "0.329281320",
      "0.760163519",
      "0.808683329",
      "0.127416348"
    ],
    [
      "0.963960003",
      "0.731872902",
      "0.565211189",
      "0.971785518"
    ],
    [
      "0.718477900",
      "0.329954910",
      "0.118952336",
      "0.820081182"
    ],
    [
      "0.909184728",
      "0.584157774",
      "0.284121306",
      "0.582748741"
    ],
    [
      "0.544840140",
      "0.967288347",
      "0.434427917",
      "0.958050614"
    ],
    [
      "0.386180421",
      "0.073260556",
      "0.489232110",
      "0.653056298"
    ],
    [
      "0.473626512",
      "0.671923158",
      "0.068130443",
      "0.489344129"
    ],
    [
      "0.072154936",
      "0.528796504",
      "0.613584611",
      "0.404534903"
    ],
    [
      "0.974216169",
      "0.628790146",
      "0.974216168",
      "0.169030165"
    ]
  ]
}

Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 16,
  "d": 4
}

The current leader's answer

{
  "points": [
    [
      "0.753297805",
      "0.908737790",
      "0.532099591",
      "0.186710510"
    ],
    [
      "0.612043453",
      "0.427460975",
      "0.390731497",
      "0.078118830"
    ],
    [
      "0.224963856",
      "0.220920131",
      "0.206494354",
      "0.240849940"
    ],
    [
      "0.532550416",
      "0.295633194",
      "0.902644984",
      "0.546220768"
    ],
    [
      "0.812443823",
      "0.123413057",
      "0.710775413",
      "0.334763234"
    ],
    [
      "0.126228403",
      "0.834022794",
      "0.321752396",
      "0.747956775"
    ],
    [
      "0.165182123",
      "0.378345307",
      "0.760038909",
      "0.902783895"
    ],
    [
      "0.329281320",
      "0.760163519",
      "0.808683329",
      "0.127416348"
    ],
    [
      "0.963960003",
      "0.731872902",
      "0.565211189",
      "0.971785518"
    ],
    [
      "0.718477900",
      "0.329954910",
      "0.118952336",
      "0.820081182"
    ],
    [
      "0.909184728",
      "0.584157774",
      "0.284121306",
      "0.582748741"
    ],
    [
      "0.544840140",
      "0.967288347",
      "0.434427917",
      "0.958050614"
    ],
    [
      "0.386180421",
      "0.073260556",
      "0.489232110",
      "0.653056298"
    ],
    [
      "0.473626512",
      "0.671923158",
      "0.068130443",
      "0.489344129"
    ],
    [
      "0.072154936",
      "0.528796504",
      "0.613584611",
      "0.404534903"
    ],
    [
      "0.974216169",
      "0.628790146",
      "0.974216168",
      "0.169030165"
    ]
  ]
}

Submit points: exactly n rows of d decimal-string coordinates in [0, 1], such as "0.25". · Verifier v1.0.0

DISCUSSION

Discussion

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