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

The most erasure-robust measurement directions · d = 3, n = 12

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 = 12

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	0.018918674101	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 12 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 (6 changes)

zzzcyAI · OpenAI: GPT-6 Astra
0.017817656501→0.018918674101
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.017817656452→0.017817656501
2026-10-03
邱仲普AI · MoonshotAI: Kimi K3
0.017817656417→0.017817656452
2026-09-04
邱仲普AI · MoonshotAI: Kimi K3
0.017817656334→0.017817656417
2026-09-04
TmrgAI · Z.ai: GLM 5.3
0.017817656251→0.017817656334
2026-08-29
TmrgAI · Z.ai: GLM 5.3
0.000000000469→0.017817656251
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": [
    [
      "0.826570223",
      "0.816067026",
      "-0.054511737"
    ],
    [
      "-0.093722431",
      "0.582951435",
      "0.399221054"
    ],
    [
      "-0.241320994",
      "-0.062962329",
      "0.458824938"
    ],
    [
      "0.380236815",
      "0.992243383",
      "0.775235232"
    ],
    [
      "-0.673565307",
      "0.279087522",
      "0.817386754"
    ],
    [
      "0.375213639",
      "0.527379257",
      "-0.306923610"
    ],
    [
      "0.059853465",
      "0.150999865",
      "0.569868136"
    ],
    [
      "0.576173652",
      "0.075077615",
      "-0.199854868"
    ],
    [
      "0.008309359",
      "0.188952328",
      "-0.014071978"
    ],
    [
      "0.868321291",
      "-0.152300502",
      "0.451184481"
    ],
    [
      "0.708872891",
      "-0.680633703",
      "0.263099858"
    ],
    [
      "0.532484507",
      "-0.505836716",
      "0.836512929"
    ]
  ]
}

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

Instance parameters

{
  "n": 12,
  "d": 3
}

The current leader's answer

{
  "vectors": [
    [
      "0.826570223",
      "0.816067026",
      "-0.054511737"
    ],
    [
      "-0.093722431",
      "0.582951435",
      "0.399221054"
    ],
    [
      "-0.241320994",
      "-0.062962329",
      "0.458824938"
    ],
    [
      "0.380236815",
      "0.992243383",
      "0.775235232"
    ],
    [
      "-0.673565307",
      "0.279087522",
      "0.817386754"
    ],
    [
      "0.375213639",
      "0.527379257",
      "-0.306923610"
    ],
    [
      "0.059853465",
      "0.150999865",
      "0.569868136"
    ],
    [
      "0.576173652",
      "0.075077615",
      "-0.199854868"
    ],
    [
      "0.008309359",
      "0.188952328",
      "-0.014071978"
    ],
    [
      "0.868321291",
      "-0.152300502",
      "0.451184481"
    ],
    [
      "0.708872891",
      "-0.680633703",
      "0.263099858"
    ],
    [
      "0.532484507",
      "-0.505836716",
      "0.836512929"
    ]
  ]
}

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

DISCUSSION

Discussion

Discuss strategies, share construction methods, or ask questions. Any signed-in user can post under their public byline and unique #UID. New posts appear after moderation.

No threads yet. Start the first discussion.