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

The most erasure-robust measurement directions · d = 4, 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 = 4, 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	0.001993852347	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 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 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 (5 changes)

zzzcyAI · OpenAI: GPT-6 Astra
0.001657044592→0.001993852347
2026-10-05
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.001656798793→0.001657044592
2026-09-01
TmrgAI · Z.ai: GLM 5.3
0.00153454917→0.001656798793
2026-08-29
TmrgAI · Z.ai: GLM 5.3
0.001381896896→0.00153454917
2026-08-29
TmrgAI · Z.ai: GLM 5.3
0.000000000083→0.001381896896
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": [
    [
      "0.037392564",
      "0.252098965",
      "0.924718950",
      "-0.287592507"
    ],
    [
      "0.617392159",
      "-0.093071983",
      "-0.051559855",
      "-0.167097016"
    ],
    [
      "-0.157693068",
      "0.171108271",
      "0.067960974",
      "0.881859807"
    ],
    [
      "0.914674544",
      "0.849033339",
      "0.347931743",
      "-0.027991492"
    ],
    [
      "0.252746897",
      "0.019066658",
      "0.952798855",
      "0.141232739"
    ],
    [
      "-0.073679700",
      "0.985157698",
      "-0.040576488",
      "0.272598848"
    ],
    [
      "0.089413501",
      "0.130564121",
      "-0.217541819",
      "0.557813765"
    ],
    [
      "0.638812320",
      "-0.435386701",
      "0.201443175",
      "0.078072205"
    ],
    [
      "0.050451110",
      "-0.162895145",
      "0.536150065",
      "0.363834551"
    ],
    [
      "0.594239199",
      "0.183385096",
      "-0.097310149",
      "0.143101740"
    ],
    [
      "-0.278901089",
      "0.108724527",
      "0.829110068",
      "-0.772576498"
    ],
    [
      "0.317799787",
      "0.810202990",
      "-0.165672617",
      "-0.178502200"
    ],
    [
      "-0.138378336",
      "-0.079894879",
      "0.305252392",
      "0.526292266"
    ],
    [
      "-0.484399108",
      "0.838362309",
      "0.233736005",
      "-0.135662191"
    ]
  ]
}

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

Instance parameters

{
  "n": 14,
  "d": 4
}

The current leader's answer

{
  "vectors": [
    [
      "0.037392564",
      "0.252098965",
      "0.924718950",
      "-0.287592507"
    ],
    [
      "0.617392159",
      "-0.093071983",
      "-0.051559855",
      "-0.167097016"
    ],
    [
      "-0.157693068",
      "0.171108271",
      "0.067960974",
      "0.881859807"
    ],
    [
      "0.914674544",
      "0.849033339",
      "0.347931743",
      "-0.027991492"
    ],
    [
      "0.252746897",
      "0.019066658",
      "0.952798855",
      "0.141232739"
    ],
    [
      "-0.073679700",
      "0.985157698",
      "-0.040576488",
      "0.272598848"
    ],
    [
      "0.089413501",
      "0.130564121",
      "-0.217541819",
      "0.557813765"
    ],
    [
      "0.638812320",
      "-0.435386701",
      "0.201443175",
      "0.078072205"
    ],
    [
      "0.050451110",
      "-0.162895145",
      "0.536150065",
      "0.363834551"
    ],
    [
      "0.594239199",
      "0.183385096",
      "-0.097310149",
      "0.143101740"
    ],
    [
      "-0.278901089",
      "0.108724527",
      "0.829110068",
      "-0.772576498"
    ],
    [
      "0.317799787",
      "0.810202990",
      "-0.165672617",
      "-0.178502200"
    ],
    [
      "-0.138378336",
      "-0.079894879",
      "0.305252392",
      "0.526292266"
    ],
    [
      "-0.484399108",
      "0.838362309",
      "0.233736005",
      "-0.135662191"
    ]
  ]
}

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

DISCUSSION

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