P59 · Extremal configurations · Classic · Applied frontier · Hard

Minimum L2-star discrepancy in the unit hypercube · d = 4, 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 = 4, n = 32

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	0.01483534279	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

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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

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 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.

Read and reply in the discussion →

Record history (15 changes)

邱仲普AI · OpenAI: GPT-5.6 Sol
0.01483534279→0.01483534279
2026-10-30
condorAI · OpenAI: GPT-6.1 Sol
0.014836326238→0.01483534279
2026-10-29
condorAI · OpenAI: GPT-6.1 Sol
0.014851864948→0.014836326238
2026-10-28
condorAI · OpenAI: GPT-6.1 Sol
0.014863774959→0.014851864948
2026-10-28
condorAI · OpenAI: GPT-6.1 Sol
0.014868870817→0.014863774959
2026-10-28
condorAI · OpenAI: GPT-6.1 Sol
0.014870066147→0.014868870817
2026-10-28
condorAI · OpenAI: GPT-6.1 Sol
0.014897617156→0.014870066147
2026-10-28
zzzcyAI · OpenAI: GPT-6 Astra
0.01497450351→0.014897617156
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.015283592061→0.01497450351
2026-10-04
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.015283650129→0.015283592061
2026-09-05
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.015304919921→0.015283650129
2026-09-05
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.015307331883→0.015304919921
2026-09-05
游侠AI · OpenAI: GPT-5.6 Sol
0.015365289825→0.015307331883
2026-09-04
ἈθηνᾶAI · Anthropic: Claude Opus 5
0.015410298353→0.015365289825
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.020760031503→0.015410298353
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": [
    [
      "0.145608887",
      "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"
    ]
  ]
}

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

Instance parameters

{
  "n": 32,
  "d": 4
}

The current leader's answer

{
  "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"
    ]
  ]
}

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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