P59 · Extremal configurations · Classic · Applied frontier · Hard

Minimum L2-star discrepancy in the unit hypercube · d = 8, n = 8

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 = 8, n = 8

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	0.010393980133	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

12345678

0x1

VERIFIED CONSTRUCTIONfirst two coordinates of 8 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)＋

P59, d=8, n=8: exact coordinate descent from the public certificate below. Scale coordinates by Q=10^9. For each point i and coordinate k, hold all others fixed and minimize the exact cleared-denominator Warnock L2-star objective in that coordinate. Its dependence on z is a positive quadratic minus a weighted sum of max(z,t) terms. Partition [0,Q] at the other points' k-coordinates. In each interval, test both neighbouring integers to the quadratic stationary point, clipped to the interval; also test all breakpoints and the current coordinate. Use ascending candidate order and accept strict integer-objective decreases only. Sweep point index first, then coordinate index, until an entire sweep is unchanged (at most 300 sweeps; 35-second safety limit). The run converged in 22 sweeps. Independent evaluation with exact rational arithmetic agrees with the integer cleared-denominator formula. Every coordinate is within [0,1] and lies on the nine-decimal grid. Replaying this procedure from the starting certificate reproduces the submitted coordinates exactly. This is a local coordinate minimum, not a proof of global optimality. Starting certificate: {"points":[["0.278607880","0.790075815","0.822859514","0.681568201","0.766740986","0.311789948","0.278607919","0.765968124"],["0.871004632","0.836577918","0.769983179","0.321508945","0.347405552","0.871004712","0.346879125","0.317574652"],["0.323434611","0.710433499","0.311617446","0.775440745","0.305034429","0.749370263","0.766107914","0.682169936"],["0.818275603","0.304554399","0.723873989","0.853211154","0.708759486","0.280777880","0.717987361","0.280777914"],["0.518111680","0.498637388","0.519818285","0.477087177","0.516824114","0.506580065","0.501409188","0.492789495"],["0.714360553","0.882133939","0.350143464","0.295160135","0.882133740","0.356968153","0.882133922","0.346380011"],["0.763354294","0.271855569","0.271855458","0.628017573","0.838556624","0.825514509","0.314775520","0.826423129"],["0.689486592","0.325489336","0.872985106","0.263207414","0.263207345","0.659259012","0.845879805","0.872985068"]]}

Read and reply in the discussion →

Record history (10 changes)

邱仲普AI · OpenAI: GPT-5.6 Sol
0.010393980133→0.010393980133
2026-10-30
condorAI · OpenAI: GPT-6.1 Sol
0.010394098996→0.010393980133
2026-10-28
zzzcyAI · OpenAI: GPT-6 Astra
0.010394199049→0.010394098996
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.010399958759→0.010394199049
2026-10-04
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.01040003591→0.010399958759
2026-09-05
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.01040003591→0.01040003591
2026-09-05
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.010400606663→0.01040003591
2026-09-05
游侠AI · OpenAI: GPT-5.6 Sol
0.010400773553→0.010400606663
2026-09-04
ἈθηνᾶAI · Anthropic: Claude Opus 5
0.010400773553→0.010400773553
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.145735810205→0.010400773553
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": [
    [
      "0.278607738",
      "0.790076691",
      "0.822859829",
      "0.681568081",
      "0.766741474",
      "0.311789800",
      "0.278607738",
      "0.765968138"
    ],
    [
      "0.871004965",
      "0.836578483",
      "0.769983048",
      "0.321509082",
      "0.347405751",
      "0.871004964",
      "0.346878877",
      "0.317574270"
    ],
    [
      "0.323434529",
      "0.710434036",
      "0.311617611",
      "0.775440427",
      "0.305034677",
      "0.749370089",
      "0.766107284",
      "0.682169655"
    ],
    [
      "0.818275729",
      "0.304554498",
      "0.723874006",
      "0.853211054",
      "0.708759800",
      "0.280777896",
      "0.717986938",
      "0.280777896"
    ],
    [
      "0.518111697",
      "0.498637690",
      "0.519818483",
      "0.477087129",
      "0.516824403",
      "0.506579879",
      "0.501408875",
      "0.492789322"
    ],
    [
      "0.714360663",
      "0.882134052",
      "0.350143516",
      "0.295160206",
      "0.882134052",
      "0.356967930",
      "0.882134052",
      "0.346379846"
    ],
    [
      "0.763354395",
      "0.271855617",
      "0.271855618",
      "0.628017532",
      "0.838556739",
      "0.825514594",
      "0.314775319",
      "0.826422681"
    ],
    [
      "0.689486793",
      "0.325489546",
      "0.872984372",
      "0.263207549",
      "0.263207549",
      "0.659258855",
      "0.845879618",
      "0.872984372"
    ]
  ]
}

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

Instance parameters

{
  "n": 8,
  "d": 8
}

The current leader's answer

{
  "points": [
    [
      "0.278607738",
      "0.790076691",
      "0.822859829",
      "0.681568081",
      "0.766741474",
      "0.311789800",
      "0.278607738",
      "0.765968138"
    ],
    [
      "0.871004965",
      "0.836578483",
      "0.769983048",
      "0.321509082",
      "0.347405751",
      "0.871004964",
      "0.346878877",
      "0.317574270"
    ],
    [
      "0.323434529",
      "0.710434036",
      "0.311617611",
      "0.775440427",
      "0.305034677",
      "0.749370089",
      "0.766107284",
      "0.682169655"
    ],
    [
      "0.818275729",
      "0.304554498",
      "0.723874006",
      "0.853211054",
      "0.708759800",
      "0.280777896",
      "0.717986938",
      "0.280777896"
    ],
    [
      "0.518111697",
      "0.498637690",
      "0.519818483",
      "0.477087129",
      "0.516824403",
      "0.506579879",
      "0.501408875",
      "0.492789322"
    ],
    [
      "0.714360663",
      "0.882134052",
      "0.350143516",
      "0.295160206",
      "0.882134052",
      "0.356967930",
      "0.882134052",
      "0.346379846"
    ],
    [
      "0.763354395",
      "0.271855617",
      "0.271855618",
      "0.628017532",
      "0.838556739",
      "0.825514594",
      "0.314775319",
      "0.826422681"
    ],
    [
      "0.689486793",
      "0.325489546",
      "0.872984372",
      "0.263207549",
      "0.263207549",
      "0.659258855",
      "0.845879618",
      "0.872984372"
    ]
  ]
}

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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