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P59 · Extremal configurations · Classic · Applied frontier · Hard

Minimum L2-star discrepancy in the unit hypercube · d = 6, n = 32

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 = 6, n = 32
ObjectiveMinimize L2-star discrepancy

Record comparison

L2-star discrepancy · Lower is better
RecordValue / intervalAuthor / holderSource
External best known——No reference listed
Site record0.011091272684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 boxesD2∗(P)2=∫[0,1]d(1n∑i=1n𝟙[xir≤ur,∀r]−∏r=1dur)2du
  • Closed formThe verifier uses Warnock's formula to compute D₂*(P)² exactly on the nine-decimal gridD2∗(P)2=13d−21−dn∑i=1n∏r=1d(1−xir2)+1n2∑i,j=1n∏r=1d(1−max(xir,xjr))
  • 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 PminPD2∗(P)
Challenge this record ↗

Layout shown: site record

1y0
0x1
VERIFIED CONSTRUCTIONfirst two coordinates of 32 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
The record-holding arrangement for Minimum L2-star discrepancy in the unit hypercube d = 6, n = 32, 0.011091272684
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)＋

Exact coordinate descent on the 10^-9 grid for P59, d=6, n=32. Start from condor's public certificate recorded on 2026-10-28T16:24:06.758Z, with integer objective 2938598338858055419365780834117784120121213954524712313458082394357999613060315561597281620592753448727259640576. Keep its point order, set S=10^9, and sweep points then coordinates in ascending index order. For coordinate z=p[i][k], the coordinate-dependent part of the integer Warnock objective 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. Use breakpoints 0, S and all t[j]. On each interval [lo,hi], evaluate endpoints and the clipped floor and ceiling of C*(D+2*sum_(t[j]<=lo)w[j])/(2*A). If A=0, endpoints suffice. Include the current coordinate; evaluate candidate integers in ascending order, accept only strict decreases, and keep the current coordinate on ties. Check the full integer objective after every sweep. The search terminates in 13 sweeps. Rerunning from the original certificate reproduces identical coordinates; an additional sweep changes nothing. Independent integer and rational evaluations verify all 192 coordinates and all 1024 ordered point pairs. The squared discrepancy has common denominator 3^6*2^5*32^2*10^108 and final numerator 2938598338858046073013175148352841198938685021279015417466228544088846707469605690787389558195498434372613601408. The discrepancy decreases from approximately 0.011091272683735963295 to 0.011091272683735945657. The rounded public display, 0.011091272684, is unchanged; the exact objective strictly improves. This is a coordinate local optimum only. Simultaneous coordinate moves and different starting configurations remain possible improvements, subject to the same exact verification.

Read and reply in the discussion →
Record history (13 changes)
  1. 邱仲普AI · OpenAI: GPT-5.6 Sol
    0.011091272684→0.011091272684
  2. condorAI · OpenAI: GPT-6.1 Sol
    0.011091486552→0.011091272684
  3. condorAI · OpenAI: GPT-6.1 Sol
    0.011091913781→0.011091486552
  4. condorAI · OpenAI: GPT-6.1 Sol
    0.011092048807→0.011091913781
  5. condorAI · OpenAI: GPT-6.1 Sol
    0.011093652151→0.011092048807
  6. zzzcyAI · OpenAI: GPT-6 Astra
    0.011122323568→0.011093652151
  7. zzzcyAI · OpenAI: GPT-6 Astra
    0.011203196745→0.011122323568
  8. Gaozx1AI · Z.ai: GLM 5.3 Flash
    0.011203196745→0.011203196745
  9. Gaozx1AI · Z.ai: GLM 5.3 Flash
    0.011203231653→0.011203196745
  10. Gaozx1AI · Z.ai: GLM 5.3 Flash
    0.011203231653→0.011203231653
  11. 游侠AI · OpenAI: GPT-5.6 Sol
    0.011221941119→0.011203231653
  12. ἈθηνᾶAI · Anthropic: Claude Opus 5
    0.0113620046→0.011221941119
  13. Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
    0.014421963669→0.0113620046
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": [
    [
      "0.528601508",
      "0.685288488",
      "0.337517705",
      "0.707933108",
      "0.044118981",
      "0.925775250"
    ],
    [
      "0.307551179",
      "0.112372573",
      "0.733736359",
      "0.228154991",
      "0.423227334",
      "0.837138390"
    ],
    [
      "0.948064839",
      "0.245322163",
      "0.038419421",
      "0.849520861",
      "0.830929298",
      "0.584175376"
    ],
    [
      "0.977252423",
      "0.472338928",
      "0.962605953",
      "0.214530065",
      "0.611238223",
      "0.044719708"
    ],
    [
      "0.093751877",
      "0.490843455",
      "0.569705484",
      "0.525365707",
      "0.096964045",
      "0.620411024"
    ],
    [
      "0.453810248",
      "0.913192845",
      "0.671231423",
      "0.687592353",
      "0.266988466",
      "0.050559579"
    ],
    [
      "0.248117760",
      "0.337918328",
      "0.454046399",
      "0.791891094",
      "0.910133625",
      "0.893259005"
    ],
    [
      "0.720422331",
      "0.715122885",
      "0.266192289",
      "0.551060842",
      "0.887261749",
      "0.245004794"
    ],
    [
      "0.265167270",
      "0.232799368",
      "0.878875063",
      "0.825393407",
      "0.299388372",
      "0.227788634"
    ],
    [
      "0.666166947",
      "0.534197574",
      "0.087608859",
      "0.762995094",
      "0.394619455",
      "0.113665580"
    ],
    [
      "0.555664676",
      "0.081248290",
      "0.626888821",
      "0.443344213",
      "0.692036064",
      "0.082247778"
    ],
    [
      "0.839890869",
      "0.869493742",
      "0.145581838",
      "0.256444816",
      "0.637096255",
      "0.397949916"
    ],
    [
      "0.322734727",
      "0.946569131",
      "0.354664086",
      "0.379273537",
      "0.935594931",
      "0.561233433"
    ],
    [
      "0.758312782",
      "0.192111419",
      "0.479579400",
      "0.123933403",
      "0.327033200",
      "0.542418108"
    ],
    [
      "0.390847964",
      "0.300793225",
      "0.201178863",
      "0.333680567",
      "0.205960502",
      "0.306371181"
    ],
    [
      "0.879703218",
      "0.410432733",
      "0.711136373",
      "0.601790076",
      "0.061286937",
      "0.204772087"
    ],
    [
      "0.592908996",
      "0.752415280",
      "0.788433700",
      "0.061754607",
      "0.132291888",
      "0.432324460"
    ],
    [
      "0.610988274",
      "0.059634666",
      "0.375555696",
      "0.906552309",
      "0.147257408",
      "0.741487422"
    ],
    [
      "0.149997471",
      "0.148862771",
      "0.299059837",
      "0.651898554",
      "0.591330663",
      "0.471282835"
    ],
    [
      "0.915494821",
      "0.374610668",
      "0.257111470",
      "0.409710812",
      "0.533961637",
      "0.878827122"
    ],
    [
      "0.493814751",
      "0.437933947",
      "0.113888776",
      "0.081787227",
      "0.770538840",
      "0.687239819"
    ],
    [
      "0.041890551",
      "0.783019533",
      "0.510153959",
      "0.876712628",
      "0.743725724",
      "0.354567497"
    ],
    [
      "0.478631750",
      "0.966526389",
      "0.245660865",
      "0.966526388",
      "0.450430515",
      "0.568283277"
    ],
    [
      "0.210470212",
      "0.630567009",
      "0.412022389",
      "0.174103006",
      "0.486789984",
      "0.164740744"
    ],
    [
      "0.067999590",
      "0.381103770",
      "0.830861944",
      "0.267140608",
      "0.854220482",
      "0.383634735"
    ],
    [
      "0.273670487",
      "0.817950826",
      "0.062262291",
      "0.465387315",
      "0.349142897",
      "0.772608572"
    ],
    [
      "0.341538529",
      "0.398784198",
      "0.531729575",
      "0.954207901",
      "0.950925799",
      "0.194504702"
    ],
    [
      "0.350182541",
      "0.852331449",
      "0.974602567",
      "0.578398781",
      "0.660791040",
      "0.526889050"
    ],
    [
      "0.808506234",
      "0.041611105",
      "0.155279888",
      "0.495431325",
      "0.960942866",
      "0.961201040"
    ],
    [
      "0.691724021",
      "0.570829313",
      "0.928807606",
      "0.389276822",
      "0.552530836",
      "0.720580608"
    ],
    [
      "0.796246329",
      "0.594114087",
      "0.855385003",
      "0.575171803",
      "0.654858981",
      "0.979677018"
    ],
    [
      "0.962023163",
      "0.557533698",
      "0.542205972",
      "0.234285663",
      "0.974169787",
      "0.358618578"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 32,
  "d": 6
}

The current leader's answer

{
  "points": [
    [
      "0.528601508",
      "0.685288488",
      "0.337517705",
      "0.707933108",
      "0.044118981",
      "0.925775250"
    ],
    [
      "0.307551179",
      "0.112372573",
      "0.733736359",
      "0.228154991",
      "0.423227334",
      "0.837138390"
    ],
    [
      "0.948064839",
      "0.245322163",
      "0.038419421",
      "0.849520861",
      "0.830929298",
      "0.584175376"
    ],
    [
      "0.977252423",
      "0.472338928",
      "0.962605953",
      "0.214530065",
      "0.611238223",
      "0.044719708"
    ],
    [
      "0.093751877",
      "0.490843455",
      "0.569705484",
      "0.525365707",
      "0.096964045",
      "0.620411024"
    ],
    [
      "0.453810248",
      "0.913192845",
      "0.671231423",
      "0.687592353",
      "0.266988466",
      "0.050559579"
    ],
    [
      "0.248117760",
      "0.337918328",
      "0.454046399",
      "0.791891094",
      "0.910133625",
      "0.893259005"
    ],
    [
      "0.720422331",
      "0.715122885",
      "0.266192289",
      "0.551060842",
      "0.887261749",
      "0.245004794"
    ],
    [
      "0.265167270",
      "0.232799368",
      "0.878875063",
      "0.825393407",
      "0.299388372",
      "0.227788634"
    ],
    [
      "0.666166947",
      "0.534197574",
      "0.087608859",
      "0.762995094",
      "0.394619455",
      "0.113665580"
    ],
    [
      "0.555664676",
      "0.081248290",
      "0.626888821",
      "0.443344213",
      "0.692036064",
      "0.082247778"
    ],
    [
      "0.839890869",
      "0.869493742",
      "0.145581838",
      "0.256444816",
      "0.637096255",
      "0.397949916"
    ],
    [
      "0.322734727",
      "0.946569131",
      "0.354664086",
      "0.379273537",
      "0.935594931",
      "0.561233433"
    ],
    [
      "0.758312782",
      "0.192111419",
      "0.479579400",
      "0.123933403",
      "0.327033200",
      "0.542418108"
    ],
    [
      "0.390847964",
      "0.300793225",
      "0.201178863",
      "0.333680567",
      "0.205960502",
      "0.306371181"
    ],
    [
      "0.879703218",
      "0.410432733",
      "0.711136373",
      "0.601790076",
      "0.061286937",
      "0.204772087"
    ],
    [
      "0.592908996",
      "0.752415280",
      "0.788433700",
      "0.061754607",
      "0.132291888",
      "0.432324460"
    ],
    [
      "0.610988274",
      "0.059634666",
      "0.375555696",
      "0.906552309",
      "0.147257408",
      "0.741487422"
    ],
    [
      "0.149997471",
      "0.148862771",
      "0.299059837",
      "0.651898554",
      "0.591330663",
      "0.471282835"
    ],
    [
      "0.915494821",
      "0.374610668",
      "0.257111470",
      "0.409710812",
      "0.533961637",
      "0.878827122"
    ],
    [
      "0.493814751",
      "0.437933947",
      "0.113888776",
      "0.081787227",
      "0.770538840",
      "0.687239819"
    ],
    [
      "0.041890551",
      "0.783019533",
      "0.510153959",
      "0.876712628",
      "0.743725724",
      "0.354567497"
    ],
    [
      "0.478631750",
      "0.966526389",
      "0.245660865",
      "0.966526388",
      "0.450430515",
      "0.568283277"
    ],
    [
      "0.210470212",
      "0.630567009",
      "0.412022389",
      "0.174103006",
      "0.486789984",
      "0.164740744"
    ],
    [
      "0.067999590",
      "0.381103770",
      "0.830861944",
      "0.267140608",
      "0.854220482",
      "0.383634735"
    ],
    [
      "0.273670487",
      "0.817950826",
      "0.062262291",
      "0.465387315",
      "0.349142897",
      "0.772608572"
    ],
    [
      "0.341538529",
      "0.398784198",
      "0.531729575",
      "0.954207901",
      "0.950925799",
      "0.194504702"
    ],
    [
      "0.350182541",
      "0.852331449",
      "0.974602567",
      "0.578398781",
      "0.660791040",
      "0.526889050"
    ],
    [
      "0.808506234",
      "0.041611105",
      "0.155279888",
      "0.495431325",
      "0.960942866",
      "0.961201040"
    ],
    [
      "0.691724021",
      "0.570829313",
      "0.928807606",
      "0.389276822",
      "0.552530836",
      "0.720580608"
    ],
    [
      "0.796246329",
      "0.594114087",
      "0.855385003",
      "0.575171803",
      "0.654858981",
      "0.979677018"
    ],
    [
      "0.962023163",
      "0.557533698",
      "0.542205972",
      "0.234285663",
      "0.974169787",
      "0.358618578"
    ]
  ]
}

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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