Matroid Correlation
Constants

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matroid #15 — Three-block sparse affine-lift GF(5) record

n
23
rank
13
bases
234,056
α(M)
15750/13283 ≈ 1.1857
status
superseded — held the GF(5) record until beaten
discovered by
Jeewon Kim
discovery date
2026-08-09
AI use
yes — GPT-5.6 Sol Ultra: AI performed the computational construction search, optimized the finite-field block and affine-lift parameters, and exhaustively verified every basis and every eligible element pair. Two independent implementations reproduced the exact correlation constant. (chat/run link)
submitted by
Jeewon Kim
submitted at
2026-08-09 14:04:27 UTC

Description (matrix) GF(5)

[[1,0,0,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0],
 [0,1,0,0,1,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0],
 [0,0,1,0,3,4,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0],
 [0,0,0,1,3,3,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0],
 [0,0,0,0,0,0,0,1,0,0,0,1,1,1,0,0,0,0,0,0,0,1,0],
 [0,0,0,0,0,0,0,0,1,0,0,1,3,0,0,0,0,0,0,0,0,2,0],
 [0,0,0,0,0,0,0,0,0,1,0,3,4,1,0,0,0,0,0,0,0,2,0],
 [0,0,0,0,0,0,0,0,0,0,1,3,3,3,0,0,0,0,0,0,0,3,0],
 [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,1,1,1,0],
 [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,3,0,2,0],
 [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,3,4,1,2,0],
 [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,3,3,3,3,0],
 [0,3,1,3,1,1,3,4,3,3,2,0,3,2,3,3,1,0,2,3,3,0,1]]

The full basis list is too large to store; recompute it from the description above.

Maximizing pair

(21, 22); basis counts at (i, j) = (21, 22):

j ∈ Bj ∉ B
i ∈ B58,800106,264
i ∉ B21,95247,040

Notes

The construction has the block form [C 0 0 u 0 0 C 0 u 0 0 0 C u 0 h1 h2 h3 0 1], where C is the rank-4, 7-column seed C = [1 0 0 0 1 1 1 0 1 0 0 1 3 0 0 0 1 0 3 4 1 0 0 0 1 3 3 3], u = [1,2,2,3]^T, h1 = [0,3,1,3,1,1,3], h2 = [4,3,3,2,0,3,2], h3 = [3,3,1,0,2,3,3]. The final two columns are the distinguished pair. The matroid has rank 13 and 23 elements. Exhaustive basis enumeration and an all-pairs check give the exact global correlation constant alpha-bar = 15750/13283, attained by the final two columns.