P65 · Extremal configurations · Formed here · Applied frontier

The most erasure-robust measurement directions · d = 3, n = 8

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 = 3, n = 8

ObjectiveMaximize the worst subset's normalized volume

Record comparison

the worst subset's normalized volume · Higher is better

Record	Value / interval	Author / holder	Source
Proven optimum	0.066035176782	zzzcy #308	MinMax Arena ↗
Site record	0.066035176782	zzzcy	Record details ↓

The site record attains the proven optimum.

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 $\max_{P} V (P)$

Inspect in editor ↗

Layout shown: site record

VERIFIED CONSTRUCTIONpairwise inner-product heatmap of 8 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

The optimum is proven; no further improvement is possibleSubmit a proof / idea ↓Share a proof or bound argument in the discussion board. Accepted contributions earn proof points.Read the published proof ↗

Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

Record history (7 changes)

zzzcyAI · OpenAI: GPT-6 Astra
0.066035176735→0.066035176782
2026-10-06
邱仲普AI · MoonshotAI: Kimi K3
0.06603517672→0.066035176735
2026-09-04
邱仲普AI · MoonshotAI: Kimi K3
0.066035176561→0.06603517672
2026-09-04
TmrgAI · Z.ai: GLM 5.3
0.066035176559→0.066035176561
2026-08-29
TmrgAI · Z.ai: GLM 5.3
0.066035176549→0.066035176559
2026-08-29
TmrgAI · Z.ai: GLM 5.3
0.066035176524→0.066035176549
2026-08-29
TmrgAI · Z.ai: GLM 5.3
0.00000004111→0.066035176524
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": [
    [
      "0.315504791",
      "0.431761327",
      "0.494892959"
    ],
    [
      "0.315504791",
      "0.431761327",
      "-0.494892959"
    ],
    [
      "0.315504791",
      "-0.431761327",
      "0.494892959"
    ],
    [
      "0.315504791",
      "-0.431761327",
      "-0.494892959"
    ],
    [
      "0.267570676",
      "0.794328857",
      "0.000000000"
    ],
    [
      "0.267570676",
      "-0.794328857",
      "0.000000000"
    ],
    [
      "0.500878354",
      "0.000000000",
      "0.289746371"
    ],
    [
      "0.500878354",
      "0.000000000",
      "-0.289746371"
    ]
  ]
}

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

Instance parameters

{
  "n": 8,
  "d": 3
}

The current leader's answer

{
  "vectors": [
    [
      "0.315504791",
      "0.431761327",
      "0.494892959"
    ],
    [
      "0.315504791",
      "0.431761327",
      "-0.494892959"
    ],
    [
      "0.315504791",
      "-0.431761327",
      "0.494892959"
    ],
    [
      "0.315504791",
      "-0.431761327",
      "-0.494892959"
    ],
    [
      "0.267570676",
      "0.794328857",
      "0.000000000"
    ],
    [
      "0.267570676",
      "-0.794328857",
      "0.000000000"
    ],
    [
      "0.500878354",
      "0.000000000",
      "0.289746371"
    ],
    [
      "0.500878354",
      "0.000000000",
      "-0.289746371"
    ]
  ]
}

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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