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

The most erasure-robust measurement directions · d = 5, n = 10

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 = 10

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	0.011149380117	zzzcy	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 10 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 holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

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 (3 changes)

zzzcyAI · OpenAI: GPT-6 Astra
0.010924461854→0.011149380117
2026-10-05
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.010924461713→0.010924461854
2026-09-01
TmrgAI · Z.ai: GLM 5.3
0.000000000011→0.010924461713
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": [
    [
      "-0.258974699",
      "0.932326143",
      "-0.472465471",
      "-0.929300964",
      "0.551186402"
    ],
    [
      "-0.201101983",
      "0.565920079",
      "0.908971091",
      "0.193513253",
      "0.354344245"
    ],
    [
      "0.748037674",
      "0.298034378",
      "-0.004576025",
      "-0.108963035",
      "-0.127444850"
    ],
    [
      "-0.360077739",
      "0.716882026",
      "0.251192589",
      "0.611807394",
      "-0.313756710"
    ],
    [
      "0.604373261",
      "0.672524773",
      "0.313504076",
      "-0.054513870",
      "0.738533166"
    ],
    [
      "0.051866742",
      "-0.316615709",
      "0.646395510",
      "-0.061519704",
      "-0.146259160"
    ],
    [
      "0.054458343",
      "-0.525260949",
      "0.339389599",
      "0.354649315",
      "0.868852024"
    ],
    [
      "-0.611197870",
      "0.174288730",
      "-0.444241345",
      "0.736043776",
      "0.470511329"
    ],
    [
      "0.976530590",
      "-0.464609995",
      "-0.605744502",
      "0.864665455",
      "0.777301312"
    ],
    [
      "0.446135580",
      "0.243230299",
      "-0.063250729",
      "0.894941376",
      "-0.683166412"
    ]
  ]
}

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

Instance parameters

{
  "n": 10,
  "d": 5
}

The current leader's answer

{
  "vectors": [
    [
      "-0.258974699",
      "0.932326143",
      "-0.472465471",
      "-0.929300964",
      "0.551186402"
    ],
    [
      "-0.201101983",
      "0.565920079",
      "0.908971091",
      "0.193513253",
      "0.354344245"
    ],
    [
      "0.748037674",
      "0.298034378",
      "-0.004576025",
      "-0.108963035",
      "-0.127444850"
    ],
    [
      "-0.360077739",
      "0.716882026",
      "0.251192589",
      "0.611807394",
      "-0.313756710"
    ],
    [
      "0.604373261",
      "0.672524773",
      "0.313504076",
      "-0.054513870",
      "0.738533166"
    ],
    [
      "0.051866742",
      "-0.316615709",
      "0.646395510",
      "-0.061519704",
      "-0.146259160"
    ],
    [
      "0.054458343",
      "-0.525260949",
      "0.339389599",
      "0.354649315",
      "0.868852024"
    ],
    [
      "-0.611197870",
      "0.174288730",
      "-0.444241345",
      "0.736043776",
      "0.470511329"
    ],
    [
      "0.976530590",
      "-0.464609995",
      "-0.605744502",
      "0.864665455",
      "0.777301312"
    ],
    [
      "0.446135580",
      "0.243230299",
      "-0.063250729",
      "0.894941376",
      "-0.683166412"
    ]
  ]
}

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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