P65 · Extremal configurations · Formed here · Applied frontier

The most erasure-robust measurement directions

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

VERIFIED CONSTRUCTIONpairwise inner-product heatmap of 10 directions (brighter = closer)

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.

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 $\max_{P} V (P)$

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

ONE LEADERBOARD PER n

Current best solutions by n

Each parameter n is an independently scored sub-problem. Select any card to inspect the current record construction or submit a stronger solution.

Discussion (0) ↓

Instanced = 3, n = 6

CURRENT RECORD(2/5)(1 − 1/√5)

Proven optimum

Record holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

Instanced = 3, n = 8

CURRENT RECORD0.066035176782

Optimal

Record holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

Instanced = 3, n = 10

CURRENT RECORD2(3 − √5)/27

Proven optimum

Record holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

Instanced = 3, n = 12

CURRENT RECORD0.018918674101

Hard

Record holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

Instanced = 4, n = 8

CURRENT RECORD0.049783309999

Proven optimum

Record holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

Instanced = 4, n = 10

CURRENT RECORD0.014086517486

Hard

Record holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

Instanced = 4, n = 12

CURRENT RECORD0.006130990397

Hard

Record holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

Instanced = 4, n = 14

CURRENT RECORD0.001993852347

Hard

Record holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

Instanced = 5, n = 10

CURRENT RECORD0.011149380117

Hard

Record holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

Instanced = 5, n = 12

CURRENT RECORD0.006479851664

Hard

Record holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

Instanced = 5, n = 14

CURRENT RECORD0.000579079584

Hard

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

Data and citation

All sub-problems in this family — authoritative scores, proof status, coordinates, and sources — are published at the stable URL below under CC BY 4.0. Because scores update as records fall, please cite the file's generatedAt timestamp.

GET https://minmaxarena.com/data/erasure-frames.json

Cite the frozen 2026-10 edition: live records change, whereas a frozen edition stays fixed and reproducible.

GET https://minmaxarena.com/data/editions/2026-10/erasure-frames.json

BibTeX (click to copy)

@misc{minmaxarena-erasure-frames-2026-10,
  title  = {{The most erasure-robust measurement directions} (P65)},
  author = {{MinMax Arena}},
  year   = {2026},
  note   = {Machine-verified records, 2026-10 edition},
  url    = {https://minmaxarena.com/data/editions/2026-10/erasure-frames.json},
  license = {CC BY 4.0}
}

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.