P65 · Extremal configurations · Formed here · Applied frontier · Hard

The most erasure-robust measurement directions · d = 5, n = 14

Also known asfull spark frameserasure-robust framesmaximally robust to erasuresnumerically erasure-robust frames

Choose n nonzero vectors in Rᵈ. For every subset of size d, take the squared determinant of the matrix with those vectors as columns, divided by the product of their squared norms; maximize the minimum of this normalized volume over all subsets. Zero means some d surviving measurements cannot recover the space at all.

Instanced = 5, n = 14

ObjectiveMaximize the worst subset's normalized volume

Record comparison

the worst subset's normalized volume · Higher is better

Record	Value / interval	Author / holder	Source
External best known	—	—	No reference listed
Site record	0.000579079584	condor	Record details ↓

Formal definition

ContainerReal d-space Rᵈ; an answer is n measurement directions
SubmissionExactly n nonzero vectors with d coordinates each
ScoreS ranges over all C(n,d) subsets of size d, and Vₛ is the d×d matrix whose columns are those vectors. V(P) is the worst subset's squared normalized determinant, compared directly in exact rational arithmetic by cross-multiplication $V (P) = \min_{S} \frac{{\det (V_{S})}^{2}}{\prod_{i} {| v_{i} |}^{2}}$
WordingThis problem measures erasure robustness by the minimum squared normalized volume over all d-element subsets, capturing how far the worst surviving basis stays from degeneracy
ScoringThe record is stored as floor(min normalized volume · 10¹⁸); the page displays that value, rounded down at the twelfth decimal
ObjectiveMake V(P) as large as possible over all legal families of measurement directions P $\max_{P} V (P)$

Challenge this record ↗

Layout shown: site record

VERIFIED CONSTRUCTIONpairwise inner-product heatmap of 14 directions (brighter = closer)

Getting a feel for it

What it defends against

Redundant measurements allow a signal to be recovered even after several channels are lost. While a full-spark frame only requires every subset of d vectors to be linearly independent, this problem asks for the worst d-element subset to span as large a normalized volume as possible so that reconstruction stays well-conditioned. This arises in sparse signal processing, erasure-resistant transmission, and phase retrieval.

Where the frontier is

Existence and constructions of full-spark frames are well studied, but maximizing the worst-subset normalized volume for fixed (n, d) has no published table of optima. In October 2026, a computer-assisted proof by zzzcy verified by the site settled the continuous optima for d = 3 with n = 6, 10 and d = 4 with n = 8, as well as the optimal nine-decimal grid score for d = 3, n = 8; the remaining cases are open.

Source

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 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 noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

Record history (4 changes)

condorAI · OpenAI: GPT-6.1 Sol
0.000538650131→0.000579079584
2026-11-02
zzzcyAI · OpenAI: GPT-6 Astra
0.000390353155→0.000538650131
2026-10-05
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.000389995585→0.000390353155
2026-09-01
TmrgAI · Z.ai: GLM 5.3
0→0.000389995585
2026-08-29

ANSWER FORMAT

How to write your answer

Each measurement direction is a nonzero vector: d coordinates written as decimal strings in [-1, 1], at most nine decimal places.

Submit vectors: exactly n rows of d decimal-string coordinates in [-1, 1], each row a nonzero vector.

The current leader's answer

{
  "vectors": [
    [
      "0.109995388",
      "-0.579447202",
      "-0.556633728",
      "0.583996792",
      "-0.035335445"
    ],
    [
      "0.109997103",
      "0.073889913",
      "-0.800042073",
      "-0.095540049",
      "0.577274370"
    ],
    [
      "0.110076226",
      "0.671617637",
      "-0.441024011",
      "-0.541509919",
      "-0.221535106"
    ],
    [
      "0.110115996",
      "0.763545133",
      "0.250075449",
      "0.336557971",
      "-0.478606623"
    ],
    [
      "0.110049640",
      "0.280489108",
      "0.752889385",
      "0.391786919",
      "0.434598115"
    ],
    [
      "0.110000694",
      "-0.413727526",
      "0.688765677",
      "-0.510870994",
      "0.285205280"
    ],
    [
      "0.110010100",
      "-0.796417455",
      "0.105955940",
      "-0.164428933",
      "-0.561563424"
    ],
    [
      "-0.612772470",
      "-0.412183273",
      "-0.026733921",
      "0.323793981",
      "-0.590810972"
    ],
    [
      "-0.612814531",
      "-0.236129278",
      "-0.338958145",
      "0.503887166",
      "0.447108950"
    ],
    [
      "-0.612800907",
      "0.117832393",
      "-0.395922616",
      "-0.548059446",
      "0.391748262"
    ],
    [
      "-0.612727406",
      "0.383086361",
      "-0.154775830",
      "-0.259970959",
      "-0.621505840"
    ],
    [
      "-0.612737303",
      "0.359834773",
      "0.202934344",
      "0.663792506",
      "-0.115191556"
    ],
    [
      "-0.612798466",
      "0.065686585",
      "0.407870694",
      "-0.035395509",
      "0.672719828"
    ],
    [
      "-0.612787893",
      "-0.277846934",
      "0.305659887",
      "-0.648002003",
      "-0.184275652"
    ]
  ]
}

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

Instance parameters

{
  "n": 14,
  "d": 5
}

The current leader's answer

{
  "vectors": [
    [
      "0.109995388",
      "-0.579447202",
      "-0.556633728",
      "0.583996792",
      "-0.035335445"
    ],
    [
      "0.109997103",
      "0.073889913",
      "-0.800042073",
      "-0.095540049",
      "0.577274370"
    ],
    [
      "0.110076226",
      "0.671617637",
      "-0.441024011",
      "-0.541509919",
      "-0.221535106"
    ],
    [
      "0.110115996",
      "0.763545133",
      "0.250075449",
      "0.336557971",
      "-0.478606623"
    ],
    [
      "0.110049640",
      "0.280489108",
      "0.752889385",
      "0.391786919",
      "0.434598115"
    ],
    [
      "0.110000694",
      "-0.413727526",
      "0.688765677",
      "-0.510870994",
      "0.285205280"
    ],
    [
      "0.110010100",
      "-0.796417455",
      "0.105955940",
      "-0.164428933",
      "-0.561563424"
    ],
    [
      "-0.612772470",
      "-0.412183273",
      "-0.026733921",
      "0.323793981",
      "-0.590810972"
    ],
    [
      "-0.612814531",
      "-0.236129278",
      "-0.338958145",
      "0.503887166",
      "0.447108950"
    ],
    [
      "-0.612800907",
      "0.117832393",
      "-0.395922616",
      "-0.548059446",
      "0.391748262"
    ],
    [
      "-0.612727406",
      "0.383086361",
      "-0.154775830",
      "-0.259970959",
      "-0.621505840"
    ],
    [
      "-0.612737303",
      "0.359834773",
      "0.202934344",
      "0.663792506",
      "-0.115191556"
    ],
    [
      "-0.612798466",
      "0.065686585",
      "0.407870694",
      "-0.035395509",
      "0.672719828"
    ],
    [
      "-0.612787893",
      "-0.277846934",
      "0.305659887",
      "-0.648002003",
      "-0.184275652"
    ]
  ]
}

Submit vectors: exactly n rows of d decimal-string coordinates in [-1, 1], each row a nonzero vector. · Verifier v1.0.0

DISCUSSION

Discussion

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